(* ::Package:: *)

(* ::Title:: *)
(*Integration problems of the form *)
(*Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n when a^2=b^2*)


(* ::Section:: *)
(*Cos[c+d x]^m (a+a Cos[c+d x])^n*)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (a+a Cos[c+d x])^n*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{Cos[a + b*x]^8, x, 4, (35*x)/128 + (35*Cos[a + b*x]*Sin[a + b*x])/(128*b) + (35*Cos[a + b*x]^3*Sin[a + b*x])/(192*b) + (7*Cos[a + b*x]^5*Sin[a + b*x])/(48*b) + (Cos[a + b*x]^7*Sin[a + b*x])/(8*b)}
{Cos[a + b*x]^7, x, 2, Sin[a + b*x]/b - Sin[a + b*x]^3/b + (3*Sin[a + b*x]^5)/(5*b) - Sin[a + b*x]^7/(7*b)}
{Cos[a + b*x]^6, x, 3, (5*x)/16 + (5*Cos[a + b*x]*Sin[a + b*x])/(16*b) + (5*Cos[a + b*x]^3*Sin[a + b*x])/(24*b) + (Cos[a + b*x]^5*Sin[a + b*x])/(6*b)}
{Cos[a + b*x]^5, x, 2, Sin[a + b*x]/b - (2*Sin[a + b*x]^3)/(3*b) + Sin[a + b*x]^5/(5*b)}
{Cos[a + b*x]^4, x, 2, (3*x)/8 + (3*Cos[a + b*x]*Sin[a + b*x])/(8*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(4*b)}
{Cos[a + b*x]^3, x, 2, Sin[a + b*x]/b - Sin[a + b*x]^3/(3*b)}
{Cos[a + b*x]^2, x, 1, x/2 + (Cos[a + b*x]*Sin[a + b*x])/(2*b)}
{Cos[a + b*x]^1, x, 1, Sin[a + b*x]/b}
{Sec[a + b*x]^1, x, 1, ArcTanh[Sin[a + b*x]]/b}
{Sec[a + b*x]^2, x, 1, Tan[a + b*x]/b}
{Sec[a + b*x]^3, x, 2, ArcTanh[Sin[a + b*x]]/(2*b) + (Sec[a + b*x]*Tan[a + b*x])/(2*b)}
{Sec[a + b*x]^4, x, 2, Tan[a + b*x]/b + Tan[a + b*x]^3/(3*b)}
{Sec[a + b*x]^5, x, 3, (3*ArcTanh[Sin[a + b*x]])/(8*b) + (3*Sec[a + b*x]*Tan[a + b*x])/(8*b) + (Sec[a + b*x]^3*Tan[a + b*x])/(4*b)}
{Sec[a + b*x]^6, x, 3, Tan[a + b*x]/b + (2*Tan[a + b*x]^3)/(3*b) + Tan[a + b*x]^5/(5*b)}
{Sec[a + b*x]^7, x, 4, (5*ArcTanh[Sin[a + b*x]])/(16*b) + (5*Sec[a + b*x]*Tan[a + b*x])/(16*b) + (5*Sec[a + b*x]^3*Tan[a + b*x])/(24*b) + (Sec[a + b*x]^5*Tan[a + b*x])/(6*b)}
{Sec[a + b*x]^8, x, 3, Tan[a + b*x]/b + Tan[a + b*x]^3/b + (3*Tan[a + b*x]^5)/(5*b) + Tan[a + b*x]^7/(7*b)}
{Sec[a + b*x]^9, x, 5, (35*ArcTanh[Sin[a + b*x]])/(128*b) + (35*Sec[a + b*x]*Tan[a + b*x])/(128*b) + (35*Sec[a + b*x]^3*Tan[a + b*x])/(192*b) + (7*Sec[a + b*x]^5*Tan[a + b*x])/(48*b) + (Sec[a + b*x]^7*Tan[a + b*x])/(8*b)}


{Cos[c + d*x]^4*(a + a*Cos[c + d*x]), x, 5, (3*a*x)/8 + (8*a*Sin[c + d*x])/(15*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (4*a*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^3*(a + a*Cos[c + d*x]), x, 4, (3*a*x)/8 + (2*a*Sin[c + d*x])/(3*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^2*(a + a*Cos[c + d*x]), x, 3, (a*x)/2 + (2*a*Sin[c + d*x])/(3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^1*(a + a*Cos[c + d*x]), x, 2, (a*x)/2 + (a*Sin[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Cos[c + d*x]^0*(a + a*Cos[c + d*x]), x, 2, a*x + (a*Sin[c + d*x])/d}
{Sec[c + d*x]^1*(a + a*Cos[c + d*x]), x, 2, a*x + (a*ArcTanh[Sin[c + d*x]])/d}
{Sec[c + d*x]^2*(a + a*Cos[c + d*x]), x, 3, (a*ArcTanh[Sin[c + d*x]])/d + (a*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(a + a*Cos[c + d*x]), x, 4, (a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(a + a*Cos[c + d*x]), x, 5, (a*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a*Tan[c + d*x])/(3*d) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(a + a*Cos[c + d*x]), x, 6, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*Tan[c + d*x])/(3*d) + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{Cos[c + d*x]^4*(a + a*Cos[c + d*x])^2, x, 6, (11*a^2*x)/16 + (16*a^2*Sin[c + d*x])/(15*d) + (11*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (8*a^2*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (11*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (2*a^2*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2, x, 5, (3*a^2*x)/4 + (6*a^2*Sin[c + d*x])/(5*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (3*a^2*Cos[c + d*x]^2*Sin[c + d*x])/(5*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(2*d) + (a^2*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2, x, 4, (7*a^2*x)/8 + (4*a^2*Sin[c + d*x])/(3*d) + (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (2*a^2*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^2, x, 3, a^2*x + (5*a^2*Sin[c + d*x])/(3*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/d + (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^2, x, 2, (3*a^2*x)/2 + (2*a^2*Sin[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^2, x, 3, 2*a^2*x + (a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Sin[c + d*x])/d}
{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^2, x, 3, a^2*x + (2*a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^2, x, 4, (3*a^2*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a^2*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^2, x, 5, (a^2*ArcTanh[Sin[c + d*x]])/d + (5*a^2*Tan[c + d*x])/(3*d) + (a^2*Sec[c + d*x]*Tan[c + d*x])/d + (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(a + a*Cos[c + d*x])^2, x, 6, (7*a^2*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^2*Tan[c + d*x])/(3*d) + (7*a^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (2*a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3, x, 6, (23*a^3*x)/16 + (34*a^3*Sin[c + d*x])/(15*d) + (23*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (17*a^3*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (23*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (13*a^3*Cos[c + d*x]^4*Sin[c + d*x])/(30*d) + (a^2*Cos[c + d*x]^4*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3, x, 5, (13*a^3*x)/8 + (13*a^3*Sin[c + d*x])/(6*d) + (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (13*a*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) - ((a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + ((a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d)}
{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^3, x, 4, (15*a^3*x)/8 + (5*a^3*Sin[c + d*x])/(2*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(4*d) + ((a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^3, x, 3, (5*a^3*x)/2 + (10*a^3*Sin[c + d*x])/(3*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (a*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^3, x, 4, (7*a^3*x)/2 + (a^3*ArcTanh[Sin[c + d*x]])/d + (5*a^3*Sin[c + d*x])/(2*d) + (a^2*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d)}
{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^3, x, 4, 3*a^3*x + (3*a^3*ArcTanh[Sin[c + d*x]])/d + (a^2*(a + a*Cos[c + d*x])*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^3, x, 4, a^3*x + (7*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*Tan[c + d*x])/(2*d) + (a^2*(a + a*Sec[c + d*x])*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^3, x, 5, (5*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (10*a^3*Tan[c + d*x])/(3*d) + (5*a^3*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (a*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(a + a*Cos[c + d*x])^3, x, 6, (15*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*Tan[c + d*x])/(2*d) + (5*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(4*d) + ((a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(a + a*Cos[c + d*x])^3, x, 7, (13*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (38*a^3*Tan[c + d*x])/(15*d) + (13*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (19*a^3*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (11*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a^2*(a + a*Cos[c + d*x])*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}


{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4, x, 6, (49*a^4*x)/16 + (49*a^4*Sin[c + d*x])/(12*d) + (49*a^4*Cos[c + d*x]*Sin[c + d*x])/(48*d) + (49*a^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(120*d) + (7*a*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(40*d) - ((a + a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + ((a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d)}
{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^4, x, 5, (7*a^4*x)/2 + (14*a^4*Sin[c + d*x])/(3*d) + (7*a^4*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (7*a^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(15*d) + (a*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d) + ((a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^4, x, 4, (35*a^4*x)/8 + (35*a^4*Sin[c + d*x])/(6*d) + (35*a^4*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (7*a^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (a*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^4, x, 5, 6*a^4*x + (a^4*ArcTanh[Sin[c + d*x]])/d + (5*a^4*Sin[c + d*x])/d + (4*a^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(3*d) + (a^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^4, x, 5, (13*a^4*x)/2 + (4*a^4*ArcTanh[Sin[c + d*x]])/d + (5*a^4*Sin[c + d*x])/(2*d) - (a^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (a^2*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^4, x, 5, 4*a^4*x + (13*a^4*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*Sin[c + d*x])/(2*d) + (3*a^3*(a + a*Cos[c + d*x])*Tan[c + d*x])/d + (a^2*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^4, x, 5, a^4*x + (6*a^4*ArcTanh[Sin[c + d*x]])/d + (5*a^4*Tan[c + d*x])/d + (4*a^3*(a + a*Sec[c + d*x])*Tan[c + d*x])/(3*d) + (a^2*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(a + a*Cos[c + d*x])^4, x, 6, (35*a^4*ArcTanh[Sin[c + d*x]])/(8*d) + (35*a^4*Tan[c + d*x])/(6*d) + (35*a^4*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (7*a^2*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (a*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(a + a*Cos[c + d*x])^4, x, 7, (7*a^4*ArcTanh[Sin[c + d*x]])/(2*d) + (14*a^4*Tan[c + d*x])/(3*d) + (7*a^4*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (7*a^2*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(15*d) + (a*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(5*d) + ((a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)}
{Sec[c + d*x]^7*(a + a*Cos[c + d*x])^4, x, 8, (49*a^4*ArcTanh[Sin[c + d*x]])/(16*d) + (24*a^4*Tan[c + d*x])/(5*d) + (49*a^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (12*a^4*Sec[c + d*x]^2*Tan[c + d*x])/(5*d) + (43*a^4*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (7*a^3*(a + a*Cos[c + d*x])*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (a^2*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^5/(a + a*Cos[c + d*x]), x, 5, (15*x)/(8*a) - (8*Sin[c + d*x])/(3*a*d) + (15*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) - (4*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - (Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^4/(a + a*Cos[c + d*x]), x, 4, -((3*x)/(2*a)) + (8*Sin[c + d*x])/(3*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + (4*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3/(a + a*Cos[c + d*x]), x, 3, (3*x)/(2*a) - (2*Sin[c + d*x])/(a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - (Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2/(a + a*Cos[c + d*x]), x, 4, -(x/a) + Sin[c + d*x]/(a*d) + Sin[c + d*x]/(a*d*(1 + Cos[c + d*x]))}
{Cos[c + d*x]^1/(a + a*Cos[c + d*x]), x, 2, x/a - Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0/(a + a*Cos[c + d*x]), x, 1, Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1/(a + a*Cos[c + d*x]), x, 3, ArcTanh[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2/(a + a*Cos[c + d*x]), x, 4, -(ArcTanh[Sin[c + d*x]]/(a*d)) + (2*Tan[c + d*x])/(a*d) - Tan[c + d*x]/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3/(a + a*Cos[c + d*x]), x, 5, (3*ArcTanh[Sin[c + d*x]])/(2*a*d) - (2*Tan[c + d*x])/(a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - (Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4/(a + a*Cos[c + d*x]), x, 6, -((3*ArcTanh[Sin[c + d*x]])/(2*a*d)) + (8*Tan[c + d*x])/(3*a*d) - (3*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + (4*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^5/(a + a*Cos[c + d*x])^2, x, 5, -((5*x)/a^2) + (8*Sin[c + d*x])/(a^2*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(a^2*d) + (4*Cos[c + d*x]^2*Sin[c + d*x])/(a^2*d) - (Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (10*Cos[c + d*x]^3*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^2, x, 4, (7*x)/(2*a^2) - (16*Sin[c + d*x])/(3*a^2*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (8*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^2, x, 4, -((2*x)/a^2) + (4*Sin[c + d*x])/(3*a^2*d) - (Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^2, x, 3, x/a^2 + Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2) - (5*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^2, x, 2, -(Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2)) + (2*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^2, x, 2, Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2) + Sin[c + d*x]/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^2, x, 3, ArcTanh[Sin[c + d*x]]/(a^2*d) - Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2) - (4*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^2, x, 5, -((2*ArcTanh[Sin[c + d*x]])/(a^2*d)) + (10*Tan[c + d*x])/(3*a^2*d) - Tan[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2) - (2*Tan[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3/(a + a*Cos[c + d*x])^2, x, 6, (7*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (16*Tan[c + d*x])/(3*a^2*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - (Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (8*Sec[c + d*x]*Tan[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4/(a + a*Cos[c + d*x])^2, x, 7, -((5*ArcTanh[Sin[c + d*x]])/(a^2*d)) + (8*Tan[c + d*x])/(a^2*d) - (5*Sec[c + d*x]*Tan[c + d*x])/(a^2*d) + (4*Sec[c + d*x]^2*Tan[c + d*x])/(a^2*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (10*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^5/(a + a*Cos[c + d*x])^3, x, 5, (13*x)/(2*a^3) - (152*Sin[c + d*x])/(15*a^3*d) + (13*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (11*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (76*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^3, x, 5, -((3*x)/a^3) + (9*Sin[c + d*x])/(5*a^3*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (3*Cos[c + d*x]^2*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) + (3*Sin[c + d*x])/(a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^3, x, 4, x/a^3 - (Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (7*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (22*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^3, x, 3, Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) - (8*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (7*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^3, x, 3, -(Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3)) + Sin[c + d*x]/(5*a*d*(a + a*Cos[c + d*x])^2) + Sin[c + d*x]/(5*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^3, x, 3, Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^3, x, 4, ArcTanh[Sin[c + d*x]]/(a^3*d) - Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) - (7*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (22*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^3, x, 6, -((3*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (24*Tan[c + d*x])/(5*a^3*d) - Tan[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) - (3*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) - (3*Tan[c + d*x])/(a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3/(a + a*Cos[c + d*x])^3, x, 7, (13*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (152*Tan[c + d*x])/(15*a^3*d) + (13*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - (Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (11*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (76*Sec[c + d*x]*Tan[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^6/(a + a*Cos[c + d*x])^4, x, 6, (21*x)/(2*a^4) - (576*Sin[c + d*x])/(35*a^4*d) + (21*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3) - (43*Cos[c + d*x]^3*Sin[c + d*x])/(35*a^2*d*(a + a*Cos[c + d*x])^2) - (288*Cos[c + d*x]^2*Sin[c + d*x])/(35*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^5/(a + a*Cos[c + d*x])^4, x, 6, -((4*x)/a^4) + (244*Sin[c + d*x])/(105*a^4*d) - (Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (12*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - (88*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + (4*Sin[c + d*x])/(a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^4, x, 5, x/a^4 - (Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Cos[c + d*x]^2*Sin[c + d*x])/(7*a*d*(a + a*Cos[c + d*x])^3) - (11*Cos[c + d*x]*Sin[c + d*x])/(21*a^2*d*(a + a*Cos[c + d*x])^2) - (32*Sin[c + d*x])/(21*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^4, x, 4, (Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + (3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - (8*Sin[c + d*x])/(35*a^2*d*(a + a*Cos[c + d*x])^2) + Sin[c + d*x]/(5*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^4, x, 4, Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) - (11*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (13*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + (13*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^4, x, 4, -(Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4)) + (4*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (8*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + (8*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^4, x, 4, Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) + (3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(35*a^2*d*(a + a*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(35*a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^4, x, 5, ArcTanh[Sin[c + d*x]]/(a^4*d) - Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) - (2*Sin[c + d*x])/(7*a*d*(a + a*Cos[c + d*x])^3) - (11*Sin[c + d*x])/(21*a^2*d*(a + a*Cos[c + d*x])^2) - (32*Sin[c + d*x])/(21*a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^4, x, 7, -((4*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (664*Tan[c + d*x])/(105*a^4*d) - Tan[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) - (12*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - (88*Tan[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - (4*Tan[c + d*x])/(a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3/(a + a*Cos[c + d*x])^4, x, 8, (21*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (576*Tan[c + d*x])/(35*a^4*d) + (21*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - (Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Sec[c + d*x]*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3) - (43*Sec[c + d*x]*Tan[c + d*x])/(35*a^2*d*(a + a*Cos[c + d*x])^2) - (288*Sec[c + d*x]*Tan[c + d*x])/(35*a^3*d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^7/(a + a*Cos[c + d*x])^5, x, 7, (31*x)/(2*a^5) - (7664*Sin[c + d*x])/(315*a^5*d) + (31*Cos[c + d*x]*Sin[c + d*x])/(2*a^5*d) - (Cos[c + d*x]^6*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (17*Cos[c + d*x]^5*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (28*Cos[c + d*x]^4*Sin[c + d*x])/(45*a^2*d*(a + a*Cos[c + d*x])^3) - (577*Cos[c + d*x]^3*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (3832*Cos[c + d*x]^2*Sin[c + d*x])/(315*a^4*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^6/(a + a*Cos[c + d*x])^5, x, 7, -((5*x)/a^5) + (181*Sin[c + d*x])/(63*a^5*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (5*Cos[c + d*x]^4*Sin[c + d*x])/(21*a*d*(a + a*Cos[c + d*x])^4) - (29*Cos[c + d*x]^3*Sin[c + d*x])/(63*a^2*d*(a + a*Cos[c + d*x])^3) - (67*Cos[c + d*x]^2*Sin[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) + (5*Sin[c + d*x])/(a^4*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^5/(a + a*Cos[c + d*x])^5, x, 6, x/a^5 - (Cos[c + d*x]^4*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (13*Cos[c + d*x]^3*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (34*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) - (173*Cos[c + d*x]*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (488*Sin[c + d*x])/(315*a^4*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^5, x, 5, (Cos[c + d*x]^4*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) + (4*Cos[c + d*x]^3*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + (4*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) - (32*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) + (4*Sin[c + d*x])/(45*a^4*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^5, x, 5, -((Cos[c + d*x]^4*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5)) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + Sin[c + d*x]/(21*a^2*d*(a + a*Cos[c + d*x])^3) - (8*Sin[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) + Sin[c + d*x]/(9*a^4*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^5, x, 5, Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) - (2*Sin[c + d*x])/(9*a*d*(a + a*Cos[c + d*x])^4) + Sin[c + d*x]/(15*a^2*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(45*a^3*d*(a + a*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(45*a^4*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^5, x, 5, -(Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5)) + (5*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + Sin[c + d*x]/(21*a^2*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(63*a^4*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^5, x, 5, Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) + (4*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + (4*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) + (8*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) + (8*Sin[c + d*x])/(315*a^4*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^5, x, 6, ArcTanh[Sin[c + d*x]]/(a^5*d) - Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) - (13*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (34*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) - (173*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (488*Sin[c + d*x])/(315*a^4*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^5, x, 8, -((5*ArcTanh[Sin[c + d*x]])/(a^5*d)) + (496*Tan[c + d*x])/(63*a^5*d) - Tan[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) - (5*Tan[c + d*x])/(21*a*d*(a + a*Cos[c + d*x])^4) - (29*Tan[c + d*x])/(63*a^2*d*(a + a*Cos[c + d*x])^3) - (67*Tan[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) - (5*Tan[c + d*x])/(a^4*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3/(a + a*Cos[c + d*x])^5, x, 9, (31*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - (7664*Tan[c + d*x])/(315*a^5*d) + (31*Sec[c + d*x]*Tan[c + d*x])/(2*a^5*d) - (Sec[c + d*x]*Tan[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (17*Sec[c + d*x]*Tan[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (28*Sec[c + d*x]*Tan[c + d*x])/(45*a^2*d*(a + a*Cos[c + d*x])^3) - (577*Sec[c + d*x]*Tan[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (3832*Sec[c + d*x]*Tan[c + d*x])/(315*a^4*d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^5/(a + a*Cos[c + d*x])^6, x, 6, (Cos[c + d*x]^5*Sin[c + d*x])/(11*d*(a + a*Cos[c + d*x])^6) + (5*Cos[c + d*x]^4*Sin[c + d*x])/(99*a*d*(a + a*Cos[c + d*x])^5) + (20*Cos[c + d*x]^3*Sin[c + d*x])/(693*a^2*d*(a + a*Cos[c + d*x])^4) + (4*Sin[c + d*x])/(231*a^3*d*(a + a*Cos[c + d*x])^3) - (32*Sin[c + d*x])/(693*a^4*d*(a + a*Cos[c + d*x])^2) + (4*Sin[c + d*x])/(99*a^5*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^6, x, 6, -((Cos[c + d*x]^5*Sin[c + d*x])/(11*d*(a + a*Cos[c + d*x])^6)) + (2*Cos[c + d*x]^4*Sin[c + d*x])/(33*a*d*(a + a*Cos[c + d*x])^5) + (8*Cos[c + d*x]^3*Sin[c + d*x])/(231*a^2*d*(a + a*Cos[c + d*x])^4) + (8*Sin[c + d*x])/(385*a^3*d*(a + a*Cos[c + d*x])^3) - (64*Sin[c + d*x])/(1155*a^4*d*(a + a*Cos[c + d*x])^2) + (8*Sin[c + d*x])/(165*a^5*d*(a + a*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (a+a Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^4*(a + a*Cos[c + d*x])^(1/2), x, 5, (32*a*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (64*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (32*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)}
{Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(1/2), x, 4, (4*a*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) - (8*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d) + (12*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)}
{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(1/2), x, 3, (14*a*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)}
{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*a*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^(1/2), x, 1, (2*a*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^(1/2), x, 1, (2*Sqrt[2]*a*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^(1/2), x, 2, (Sqrt[2]*a*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^(1/2), x, 3, (3*a*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (3*a*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^(1/2), x, 4, (5*a*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (5*a*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (5*a*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2), x, 6, (68*a^2*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (34*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (136*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (68*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d)}
{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2), x, 4, (152*a^2*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (38*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (4*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)}
{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^(3/2), x, 3, (8*a^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^(3/2), x, 2, (8*a^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^(3/2), x, 3, (2*Sqrt[2]*a^2*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2), x, 3, (3*Sqrt[2]*a^2*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2), x, 4, (7*a^2*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (7*a^2*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^(3/2), x, 5, (11*a^2*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (11*a^2*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (11*a^2*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2), x, 6, (284*a^3*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) + (710*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (46*a^3*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (568*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a^2*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (284*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(231*d)}
{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2), x, 5, (832*a^3*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (208*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (26*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) - (4*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)}
{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^(5/2), x, 4, (64*a^3*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^(5/2), x, 3, (64*a^3*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^(5/2), x, 3, (2*Sqrt[2]*a^3*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (14*a^3*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2), x, 3, (5*Sqrt[2]*a^3*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2), x, 3, (19*a^3*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (9*a^3*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^(5/2), x, 4, (25*a^3*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (25*a^3*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (13*a^3*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(a + a*Cos[c + d*x])^(5/2), x, 5, (163*a^3*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(32*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (163*a^3*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (163*a^3*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (17*a^3*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{(a + a*Cos[c + d*x])^(7/2), x, 4, (256*a^4*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (64*a^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d) + (24*a^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*a*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^(1/2), x, 5, (2*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) - (148*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (62*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)}
{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^(1/2), x, 4, -((2*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + (28*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) - (2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)}
{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^(1/2), x, 3, (2*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) - (4*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)}
{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^(1/2), x, 2, -((2*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + (2*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^(1/2), x, 1, (2*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^1/Sqrt[a + a*Cos[c + d*x]], x, 3, (-2*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^2/Sqrt[a + a*Cos[c + d*x]], x, 4, (2*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) - (Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + Tan[c + d*x]/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^3/Sqrt[a + a*Cos[c + d*x]], x, 5, (-2*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (7*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - Tan[c + d*x]/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^4/Sqrt[a + a*Cos[c + d*x]], x, 6, (2*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) - (9*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (7*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}


{Cos[x]/Sqrt[a + a*Cos[x]], x, 2, -((2*ArcTanh[Sin[x/2]]*Cos[x/2])/Sqrt[a + a*Cos[x]]) + (2*Sin[x])/Sqrt[a + a*Cos[x]]}
{Sec[x]/Sqrt[a + a*Cos[x]], x, 3, (-2*ArcTanh[Sin[x/2]]*Cos[x/2])/Sqrt[a + a*Cos[x]] + (2*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[x/2]]*Cos[x/2])/Sqrt[a + a*Cos[x]]}


{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^(3/2), x, 5, -((15*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])) - (Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (31*Sin[c + d*x])/(5*a*d*Sqrt[a + a*Cos[c + d*x]]) + (9*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - (13*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d)}
{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^(3/2), x, 4, (11*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - (Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - (13*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) + (7*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)}
{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^(3/2), x, 3, -((7*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])) + Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^(3/2), x, 2, (3*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^(3/2), x, 2, (ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) + Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^(3/2), x, 4, (-5*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(a*d*Sqrt[a + a*Cos[c + d*x]]) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^(3/2), x, 5, (9*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - (3*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(a*d*Sqrt[a + a*Cos[c + d*x]]) - Tan[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (3*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^3/(a + a*Cos[c + d*x])^(3/2), x, 6, (-13*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) + (19*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*a*d*Sqrt[a + a*Cos[c + d*x]]) - (7*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (Sec[c + d*x]*Tan[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^(5/2), x, 5, (163*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (17*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - (197*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + (95*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)}
{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^(5/2), x, 4, -((75*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])) - (Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (13*Cos[c + d*x]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (49*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^(5/2), x, 3, (19*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (13*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^(5/2), x, 3, (5*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (5*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^(5/2), x, 3, (3*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^(5/2), x, 5, (-43*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(a^2*d*Sqrt[a + a*Cos[c + d*x]]) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (11*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^(5/2), x, 6, (115*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - (5*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(a^2*d*Sqrt[a + a*Cos[c + d*x]]) - Tan[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (15*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (35*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (a+a Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{Cos[a + b*x]^(11/2), x, 4, (30*EllipticF[(1/2)*(a + b*x), 2])/(77*b) + (30*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(77*b) + (18*Cos[a + b*x]^(5/2)*Sin[a + b*x])/(77*b) + (2*Cos[a + b*x]^(9/2)*Sin[a + b*x])/(11*b)}
{Cos[a + b*x]^(9/2), x, 3, (14*EllipticE[(1/2)*(a + b*x), 2])/(15*b) + (14*Cos[a + b*x]^(3/2)*Sin[a + b*x])/(45*b) + (2*Cos[a + b*x]^(7/2)*Sin[a + b*x])/(9*b)}
{Cos[a + b*x]^(7/2), x, 3, (10*EllipticF[(1/2)*(a + b*x), 2])/(21*b) + (10*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(21*b) + (2*Cos[a + b*x]^(5/2)*Sin[a + b*x])/(7*b)}
{Cos[a + b*x]^(5/2), x, 2, (6*EllipticE[(1/2)*(a + b*x), 2])/(5*b) + (2*Cos[a + b*x]^(3/2)*Sin[a + b*x])/(5*b)}
{Cos[a + b*x]^(3/2), x, 2, (2*EllipticF[(1/2)*(a + b*x), 2])/(3*b) + (2*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(3*b)}
{Cos[a + b*x]^(1/2), x, 1, (2*EllipticE[(1/2)*(a + b*x), 2])/b}
{1/Cos[a + b*x]^(1/2), x, 1, (2*EllipticF[(1/2)*(a + b*x), 2])/b}
{1/Cos[a + b*x]^(3/2), x, 2, -((2*EllipticE[(1/2)*(a + b*x), 2])/b) + (2*Sin[a + b*x])/(b*Sqrt[Cos[a + b*x]])}
{1/Cos[a + b*x]^(5/2), x, 2, (2*EllipticF[(1/2)*(a + b*x), 2])/(3*b) + (2*Sin[a + b*x])/(3*b*Cos[a + b*x]^(3/2))}
{1/Cos[a + b*x]^(7/2), x, 3, -((6*EllipticE[(1/2)*(a + b*x), 2])/(5*b)) + (2*Sin[a + b*x])/(5*b*Cos[a + b*x]^(5/2)) + (6*Sin[a + b*x])/(5*b*Sqrt[Cos[a + b*x]])}
{1/Cos[a + b*x]^(9/2), x, 3, (10*EllipticF[(1/2)*(a + b*x), 2])/(21*b) + (2*Sin[a + b*x])/(7*b*Cos[a + b*x]^(7/2)) + (10*Sin[a + b*x])/(21*b*Cos[a + b*x]^(3/2))}
{1/Cos[a + b*x]^(11/2), x, 4, -((14*EllipticE[(1/2)*(a + b*x), 2])/(15*b)) + (2*Sin[a + b*x])/(9*b*Cos[a + b*x]^(9/2)) + (14*Sin[a + b*x])/(45*b*Cos[a + b*x]^(5/2)) + (14*Sin[a + b*x])/(15*b*Sqrt[Cos[a + b*x]])}


{Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]), x, 6, (6*a*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (10*a*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]), x, 5, (6*a*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x]), x, 4, (2*a*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x]), x, 3, (2*a*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*EllipticF[(1/2)*(c + d*x), 2])/d}
{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x]), x, 4, -((2*a*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x]), x, 5, -((2*a*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x]), x, 6, -((6*a*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*a*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])}


{Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2, x, 7, (32*a^2*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (20*a^2*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (20*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (32*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (4*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^2*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2, x, 6, (12*a^2*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (8*a^2*EllipticF[(1/2)*(c + d*x), 2])/(7*d) + (8*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (4*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^2, x, 5, (16*a^2*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^2*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (4*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^2, x, 4, (4*a^2*EllipticE[(1/2)*(c + d*x), 2])/d + (8*a^2*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^2, x, 3, (4*a^2*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^2, x, 5, -((4*a^2*EllipticE[(1/2)*(c + d*x), 2])/d) + (8*a^2*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^2, x, 6, -((16*a^2*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^2*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (16*a^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])}


{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3, x, 7, (68*a^3*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (44*a^3*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (44*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (68*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (40*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*a^2*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^3, x, 6, (28*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (52*a^3*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (52*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (32*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*a^2*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^3, x, 5, (36*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^3*EllipticF[(1/2)*(c + d*x), 2])/d + (8*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^3, x, 5, (4*a^3*EllipticE[(1/2)*(c + d*x), 2])/d + (20*a^3*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (4*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(a + a*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^3, x, 5, -((4*a^3*EllipticE[(1/2)*(c + d*x), 2])/d) + (20*a^3*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (16*a^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + a*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^3, x, 6, -((36*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^3*EllipticF[(1/2)*(c + d*x), 2])/d + (8*a^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (36*a^3*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(a + a*Cos[c + d*x])^3, x, 7, -((28*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (52*a^3*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (32*a^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (52*a^3*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (28*a^3*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}


{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^4, x, 8, (128*a^4*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (904*a^4*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (904*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (128*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (860*a^4*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (52*a^3*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(99*d) + (2*a^2*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^4, x, 7, (152*a^4*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (32*a^4*EllipticF[(1/2)*(c + d*x), 2])/(7*d) + (32*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (188*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (44*a^3*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(63*d) + (2*a^2*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^4, x, 6, (64*a^4*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (136*a^4*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (332*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (36*a^3*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d) + (2*a^2*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^4, x, 6, (56*a^4*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (32*a^4*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (4*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (8*a^3*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (2*a^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^4, x, 5, (40*a^4*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (20*a^4*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^4, x, 6, -((56*a^4*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (32*a^4*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (164*a^4*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (28*a^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(a + a*Cos[c + d*x])^4, x, 7, -((64*a^4*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (136*a^4*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (332*a^4*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (64*a^4*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (36*a^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*a^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x]), x, 6, (21*EllipticE[(1/2)*(c + d*x), 2])/(5*a*d) - (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + (7*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x]), x, 5, -((3*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) + (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x]), x, 4, (3*EllipticE[(1/2)*(c + d*x), 2])/(a*d) - EllipticF[(1/2)*(c + d*x), 2]/(a*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x]), x, 4, -(EllipticE[(1/2)*(c + d*x), 2]/(a*d)) + EllipticF[(1/2)*(c + d*x), 2]/(a*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x]), x, 4, EllipticE[(1/2)*(c + d*x), 2]/(a*d) + EllipticF[(1/2)*(c + d*x), 2]/(a*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x]), x, 5, -((3*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - EllipticF[(1/2)*(c + d*x), 2]/(a*d) + (3*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x]), x, 6, (3*EllipticE[(1/2)*(c + d*x), 2])/(a*d) + (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (5*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (3*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^2, x, 7, (56*EllipticE[(1/2)*(c + d*x), 2])/(5*a^2*d) - (5*EllipticF[(1/2)*(c + d*x), 2])/(a^2*d) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d) + (56*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^2, x, 6, -((7*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + (10*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (10*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (7*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^2, x, 5, (4*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) - (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^2, x, 5, -(EllipticE[(1/2)*(c + d*x), 2]/(a^2*d)) + (2*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^2, x, 2, EllipticF[(1/2)*(c + d*x), 2]/(3*a^2*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)}
{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^2, x, 5, EllipticE[(1/2)*(c + d*x), 2]/(a^2*d) + (2*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^2, x, 6, -((4*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) - (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (4*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - (5*Sin[c + d*x])/(3*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^2, x, 7, (7*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (10*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (10*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - (7*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - (7*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^(11/2)/(a + a*Cos[c + d*x])^3, x, 8, (231*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) - (21*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) - (21*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) + (77*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a^3*d) - (Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (4*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) - (63*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^3, x, 7, -((119*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + (11*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) + (11*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - (119*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^3, x, 6, (49*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) - (13*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (8*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (13*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^3, x, 6, -((9*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + EllipticF[(1/2)*(c + d*x), 2]/(2*a^3*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) + (9*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^3, x, 6, -(EllipticE[(1/2)*(c + d*x), 2]/(10*a^3*d)) + EllipticF[(1/2)*(c + d*x), 2]/(6*a^3*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^3, x, 6, EllipticE[(1/2)*(c + d*x), 2]/(10*a^3*d) + EllipticF[(1/2)*(c + d*x), 2]/(6*a^3*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^3, x, 6, (9*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + EllipticF[(1/2)*(c + d*x), 2]/(2*a^3*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) - (9*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^3, x, 7, -((49*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) - (13*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + (49*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - (8*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - (13*Sin[c + d*x])/(6*a^2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^3, x, 8, (119*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + (11*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) + (11*Sin[c + d*x])/(2*a^3*d*Cos[c + d*x]^(3/2)) - (119*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - (2*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - (119*Sin[c + d*x])/(30*a^2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (a+a Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(1/2), x, 4, (5*Sqrt[a]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) + (5*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (5*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(1/2), x, 3, (3*Sqrt[a]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) + (3*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^(1/2), x, 2, (Sqrt[a]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^(1/2), x, 1, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d}
{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^(1/2), x, 1, (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-9/2)*(a + a*Cos[c + d*x])^(1/2), x, 4, (2*a*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (12*a*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Sin[c + d*x])/(35*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (32*a*Sin[c + d*x])/(35*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2), x, 5, (11*a^(3/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) + (11*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (11*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (7*a^(3/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) + (7*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^(3/2), x, 3, (3*a^(3/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^(3/2), x, 3, (2*a^(3/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^(3/2), x, 2, (8*a^2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^(3/2), x, 3, (8*a^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(a + a*Cos[c + d*x])^(3/2), x, 5, (2*a^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (26*a^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (104*a^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (208*a^2*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (163*a^(5/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*d) + (163*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (163*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (17*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^(5/2), x, 4, (25*a^(5/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) + (25*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (13*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^(5/2), x, 3, (19*a^(5/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) + (9*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)}
{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^(5/2), x, 3, (5*a^(5/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d - (a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^(5/2), x, 3, (2*a^(5/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (14*a^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^(5/2), x, 3, (64*a^3*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(a + a*Cos[c + d*x])^(5/2), x, 4, (64*a^3*Sin[c + d*x])/(21*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2)) + (2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{Cos[c + d*x]^(-11/2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (38*a^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (146*a^3*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (584*a^3*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (1168*a^3*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}


{Cos[c + d*x]^(-5/4)*(a + a*Cos[c + d*x])^(3/2), x, 2, (4*a^2*Sin[c + d*x])/(d*Cos[c + d*x]^(1/4)*Sqrt[a + a*Cos[c + d*x]])}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(5/2)/(1 + Cos[c + d*x])^(1/2), x, 5, -((Sqrt[2]*ArcSin[Tan[(1/2)*(c + d*x)]])/d) + (7*ArcTan[Sin[c + d*x]/(Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])])/(4*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[1 + Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[1 + Cos[c + d*x]])}
{Cos[c + d*x]^(3/2)/(1 + Cos[c + d*x])^(1/2), x, 4, (Sqrt[2]*ArcSin[Tan[(1/2)*(c + d*x)]])/d - ArcTan[Sin[c + d*x]/(Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]/d + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]])}
{Cos[c + d*x]^(1/2)/(1 + Cos[c + d*x])^(1/2), x, 3, -((Sqrt[2]*ArcSin[Tan[(1/2)*(c + d*x)]])/d) + (2*ArcTan[Sin[c + d*x]/(Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])])/d}
{Cos[c + d*x]^(-1/2)/(1 + Cos[c + d*x])^(1/2), x, 1, (Sqrt[2]*ArcSin[Tan[(1/2)*(c + d*x)]])/d}
{Cos[c + d*x]^(-3/2)/(1 + Cos[c + d*x])^(1/2), x, 2, -((Sqrt[2]*ArcSin[Tan[(1/2)*(c + d*x)]])/d) + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)/(1 + Cos[c + d*x])^(1/2), x, 3, (Sqrt[2]*ArcSin[Tan[(1/2)*(c + d*x)]])/d + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[1 + Cos[c + d*x]]) - (2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)/(1 + Cos[c + d*x])^(1/2), x, 4, -((Sqrt[2]*ArcSin[Tan[(1/2)*(c + d*x)]])/d) + (2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[1 + Cos[c + d*x]]) - (2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[1 + Cos[c + d*x]]) + (26*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])}


{1/(Sqrt[1 + Cos[x]]*Sqrt[Cos[x]]), x, 1, Sqrt[2]*ArcSin[Tan[x/2]]}
{1/(Sqrt[a + a*Cos[x]]*Sqrt[Cos[x]]), x, 1, (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[Cos[x]]*Sqrt[a + a*Cos[x]])])/Sqrt[a]}


{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(1/2), x, 5, (7*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(1/2), x, 4, -(ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]/(Sqrt[a]*d)) + (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(1/2), x, 3, (2*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)}
{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^(1/2), x, 1, (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)}
{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^(1/2), x, 2, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^(1/2), x, 3, (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)/(a + a*Cos[c + d*x])^(1/2), x, 4, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (26*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(3/2), x, 5, -((3*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(3/2)*d)) + (9*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(3/2), x, 4, (2*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(3/2)*d) - (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(3/2), x, 2, ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^(3/2), x, 2, (3*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^(3/2), x, 3, -((7*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - Sin[c + d*x]/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + (5*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^(3/2), x, 4, (11*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (7*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (19*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(5/2), x, 6, -((5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(5/2)*d)) + (115*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (15*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (35*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(5/2), x, 5, (2*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(5/2)*d) - (43*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (11*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(5/2), x, 3, (3*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(5/2), x, 3, (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^(5/2), x, 3, (19*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (9*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^(5/2), x, 4, -((75*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - Sin[c + d*x]/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - (13*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + (49*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^(5/2), x, 5, (163*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - (17*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (95*Sin[c + d*x])/(48*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (299*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^(7/2), x, 7, -((7*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(7/2)*d)) + (637*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (7*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - (259*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + (189*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(7/2), x, 6, (2*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(7/2)*d) - (177*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (17*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - (49*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(7/2), x, 4, (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + (5*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) + (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(7/2), x, 4, (7*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + (7*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) + (7*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(7/2), x, 4, (13*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^(7/2), x, 4, (63*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - (103*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^(7/2), x, 5, -((363*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d)) - Sin[c + d*x]/(6*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)) - (19*Sin[c + d*x])/(48*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - (199*Sin[c + d*x])/(192*a^2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + (691*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^(7/2), x, 6, (1015*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - Sin[c + d*x]/(6*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)) - (23*Sin[c + d*x])/(48*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - (109*Sin[c + d*x])/(64*a^2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (193*Sin[c + d*x])/(64*a^3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (629*Sin[c + d*x])/(64*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(9/2), x, 5, (35*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(1024*Sqrt[2]*a^(9/2)*d) + (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(8*d*(a + a*Cos[c + d*x])^(9/2)) + (7*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(96*a*d*(a + a*Cos[c + d*x])^(7/2)) + (35*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(768*a^2*d*(a + a*Cos[c + d*x])^(5/2)) + (35*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(1024*a^3*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(9/2), x, 5, (45*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(1024*Sqrt[2]*a^(9/2)*d) - (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(8*d*(a + a*Cos[c + d*x])^(9/2)) + (3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(32*a*d*(a + a*Cos[c + d*x])^(7/2)) + (15*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(256*a^2*d*(a + a*Cos[c + d*x])^(5/2)) + (45*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(1024*a^3*d*(a + a*Cos[c + d*x])^(3/2))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (a - a Cos[c+d x])^(n/2)*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^(3/2)*(a - a*Cos[c + d*x])^(1/2), x, 3, (3*Sqrt[-a]*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(4*d) + (3*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a - a*Cos[c + d*x]]) - (a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a - a*Cos[c + d*x]])}
{Cos[c + d*x]^(1/2)*(a - a*Cos[c + d*x])^(1/2), x, 2, -((Sqrt[-a]*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/d) - (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a - a*Cos[c + d*x]])}
{Cos[c + d*x]^(-1/2)*(a - a*Cos[c + d*x])^(1/2), x, 1, (2*Sqrt[-a]*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/d}
{Cos[c + d*x]^(-3/2)*(a - a*Cos[c + d*x])^(1/2), x, 1, (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(a - a*Cos[c + d*x])^(1/2), x, 2, (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) - (4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(a - a*Cos[c + d*x])^(1/2), x, 3, (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a - a*Cos[c + d*x]]) - (8*a*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) + (16*a*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])}


{Cos[c + d*x]^(3/2)*(1 - Cos[c + d*x])^(1/2), x, 3, -((3*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/(4*d)) + (3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[1 - Cos[c + d*x]]) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[1 - Cos[c + d*x]])}
{Cos[c + d*x]^(1/2)*(1 - Cos[c + d*x])^(1/2), x, 2, ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])]/d - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]])}
{Cos[c + d*x]^(-1/2)*(1 - Cos[c + d*x])^(1/2), x, 1, -((2*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d)}
{Cos[c + d*x]^(-3/2)*(1 - Cos[c + d*x])^(1/2), x, 1, (2*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(1 - Cos[c + d*x])^(1/2), x, 2, (2*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)) - (4*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(1 - Cos[c + d*x])^(1/2), x, 3, (2*Sin[c + d*x])/(5*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(5/2)) - (8*Sin[c + d*x])/(15*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)) + (16*Sin[c + d*x])/(15*d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(5/2)/(1 - Cos[c + d*x])^(1/2), x, 5, (7*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/(4*d) - (Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[1 - Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[1 - Cos[c + d*x]])}
{Cos[c + d*x]^(3/2)/(1 - Cos[c + d*x])^(1/2), x, 4, ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])]/d - (Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]])}
{Cos[c + d*x]^(1/2)/(1 - Cos[c + d*x])^(1/2), x, 3, (2*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d - (Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d}
{Cos[c + d*x]^(-1/2)/(1 - Cos[c + d*x])^(1/2), x, 1, -((Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d)}
{Cos[c + d*x]^(-3/2)/(1 - Cos[c + d*x])^(1/2), x, 2, -((Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d) + (2*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)/(1 - Cos[c + d*x])^(1/2), x, 3, -((Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d) + (2*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)) + (2*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])}


{1/(Sqrt[1 - Cos[x]]*Sqrt[Cos[x]]), x, 1, (-Sqrt[2])*ArcTanh[Sin[x]/(Sqrt[2]*Sqrt[1 - Cos[x]]*Sqrt[Cos[x]])]}
{1/(Sqrt[a - a*Cos[x]]*Sqrt[Cos[x]]), x, 1, -((Sqrt[2]*ArcTan[(Sqrt[-a]*Sin[x])/(Sqrt[2]*Sqrt[Cos[x]]*Sqrt[a - a*Cos[x]])])/Sqrt[-a])}


{Cos[c + d*x]^(5/2)/(a - a*Cos[c + d*x])^(1/2), x, 5, (7*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(4*Sqrt[-a]*d) - (Sqrt[2]*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[-a]*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a - a*Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a - a*Cos[c + d*x]])}
{Cos[c + d*x]^(3/2)/(a - a*Cos[c + d*x])^(1/2), x, 4, ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])]/(Sqrt[-a]*d) - (Sqrt[2]*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[-a]*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a - a*Cos[c + d*x]])}
{Cos[c + d*x]^(1/2)/(a - a*Cos[c + d*x])^(1/2), x, 3, (2*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[-a]*d) - (Sqrt[2]*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[-a]*d)}
{Cos[c + d*x]^(-1/2)/(a - a*Cos[c + d*x])^(1/2), x, 1, -((Sqrt[2]*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[-a]*d))}
{Cos[c + d*x]^(-3/2)/(a - a*Cos[c + d*x])^(1/2), x, 2, -((Sqrt[2]*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[-a]*d)) + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)/(a - a*Cos[c + d*x])^(1/2), x, 3, -((Sqrt[2]*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[-a]*d)) + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) + (2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)/(a - a*Cos[c + d*x])^(1/2), x, 4, -((Sqrt[2]*ArcTan[(Sqrt[-a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[-a]*d)) + (2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a - a*Cos[c + d*x]]) + (2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) + (26*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/3) (a+a Cos[c+d x])^(n/3)*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(1/3), x, 2, Int[Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(1/3), x]}
{Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(2/3), x, 2, Int[((5*a^2)/9 + (7/9)*a^2*Cos[c + d*x])/(Cos[c + d*x]^(2/3)*(a + a*Cos[c + d*x])^(1/3)), x]/(2*a) + (5*a*Cos[c + d*x]^(1/3)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(1/3)) + (a*Cos[c + d*x]^(4/3)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(1/3))}
{Cos[c + d*x]^(5/3)*(a + a*Cos[c + d*x])^(2/3), x, 2, (9*Int[((4*a^2)/3 + (14/9)*a^2*Cos[c + d*x])/(Cos[c + d*x]^(1/3)*(a + a*Cos[c + d*x])^(1/3)), x])/(28*a) + (9*a*Cos[c + d*x]^(2/3)*Sin[c + d*x])/(14*d*(a + a*Cos[c + d*x])^(1/3)) + (3*a*Cos[c + d*x]^(5/3)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^(1/3))}


(* ::Subsubsection:: *)
(*n<0*)


(* ::Subsection::Closed:: *)
(*Integrands of the form Sec[c+d x]^(m/2) (a+a Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 7, -((6*a*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 6, (-2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 5, (-2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])*Sec[c + d*x]^(1/2), x, 4, (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d}
{(a + a*Cos[c + d*x])/Sec[c + d*x]^(1/2), x, 5, (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])/Sec[c + d*x]^(3/2), x, 6, (6*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])/Sec[c + d*x]^(5/2), x, 7, (6*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (10*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (10*a*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2), x, 7, -((16*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (16*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2), x, 6, (-4*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2), x, 4, (4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(1/2), x, 5, (4*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^2/Sec[c + d*x]^(1/2), x, 6, (16*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (4*a^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^2/Sec[c + d*x]^(3/2), x, 7, (12*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(7*d) + (2*a^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (4*a^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (8*a^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2), x, 8, -((28*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (52*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (28*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (52*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (32*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a^2*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2), x, 7, -((36*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (36*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^2*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2), x, 6, -((4*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (20*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (16*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2), x, 6, (4*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (20*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(1/2), x, 6, (36*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^3*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^3/Sec[c + d*x]^(1/2), x, 7, (28*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (52*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (32*a^3*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*a^2*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(3/2)) + (52*a^3*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^3/Sec[c + d*x]^(3/2), x, 8, (68*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (44*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (40*a^3*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*a^2*(a + a*Cos[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^(5/2)) + (68*a^3*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (44*a^3*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(9/2), x, 8, -((64*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (136*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (64*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (332*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (36*a^3*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a^2*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(7/2), x, 7, -((56*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (164*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (28*a^3*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a^2*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(5/2), x, 6, (40*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (20*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(3/2), x, 7, (56*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^4*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (8*a^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(1/2), x, 7, (64*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (136*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (332*a^4*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (36*a^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^4/Sec[c + d*x]^(1/2), x, 8, (152*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(7*d) + (188*a^4*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (44*a^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(3/2)) + (2*a^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (32*a^4*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x]), x, 7, (3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - (3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + (5*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x]), x, 6, (-3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x]), x, 5, (Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - Sin[c + d*x]/(d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(1/2)), x, 5, -((Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + Sin[c + d*x]/(d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 5, (3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - Sin[c + d*x]/(d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)), x, 6, (-3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - Sin[c + d*x]/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + (5*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)), x, 7, (21*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - Sin[c + d*x]/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (7*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (5*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])}


{Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^2, x, 8, (7*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (7*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (10*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (7*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^2, x, 7, (-4*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (5*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^2, x, 6, (Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(a*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(1/2)), x, 3, (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)), x, 6, -((Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) + Sin[c + d*x]/(a*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)), x, 6, (4*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) - (5*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)), x, 7, -((7*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - (7*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + (10*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)), x, 8, (56*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - (3*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (56*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*Sin[c + d*x])/(a^2*d*Sqrt[Sec[c + d*x]])}


{Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^3, x, 9, (119*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (11*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - (119*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + (11*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a^3*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - (119*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^3, x, 8, (-49*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - (13*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (49*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (8*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (13*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^3, x, 7, (9*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]) - (2*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - (9*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(1/2)), x, 7, (Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]) + Sin[c + d*x]/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(10*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)), x, 7, -(Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]) + (4*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) + Sin[c + d*x]/(10*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)), x, 7, (-9*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)) - (2*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) + (9*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)), x, 7, (49*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - (13*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)) - (8*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) - (13*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)), x, 8, -((119*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) + (11*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)) - (2*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - (119*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + (11*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(11/2)), x, 9, (231*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - (21*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)) - (4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - (63*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (77*Sin[c + d*x])/(10*a^3*d*Sec[c + d*x]^(3/2)) - (21*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Sec[c+d x]^(m/2) (a+a Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2), x, 4, (32*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (12*a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2), x, 3, (16*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2), x, 2, (4*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2), x, 1, (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(1/2), x, 2, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d}
{Sqrt[a + a*Cos[c + d*x]]/Sec[c + d*x]^(1/2), x, 3, (Sqrt[a]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]/Sec[c + d*x]^(3/2), x, 4, (3*Sqrt[a]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (3*a*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2), x, 5, (208*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (104*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (26*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])}
{(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2), x, 3, (8*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2), x, 2, (8*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2), x, 4, (2*a^(3/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(1/2), x, 4, (3*a^(3/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^2*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^(3/2)/Sec[c + d*x]^(1/2), x, 5, (7*a^(3/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (7*a^2*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^(3/2)/Sec[c + d*x]^(3/2), x, 6, (11*a^(3/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (11*a^2*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (11*a^2*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2), x, 5, (1168*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (584*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (146*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (38*a^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2), x, 4, (64*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2), x, 3, (64*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2), x, 4, (2*a^(5/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (14*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2), x, 4, (5*a^(5/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^3*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(1/2), x, 4, (19*a^(5/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (9*a^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^(5/2)/Sec[c + d*x]^(1/2), x, 5, (25*a^(5/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (13*a^3*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (25*a^3*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^(5/2)/Sec[c + d*x]^(3/2), x, 6, (163*a^(5/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (17*a^3*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (163*a^3*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (163*a^3*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Sec[c + d*x]^(7/2)/Sqrt[1 + Cos[c + d*x]], x, 5, -((Sqrt[2]*ArcSin[Tan[(1/2)*(c + d*x)]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d) + (26*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[1 + Cos[c + d*x]]) - (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[1 + Cos[c + d*x]]) + (2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[1 + Cos[c + d*x]])}
{Sec[c + d*x]^(5/2)/Sqrt[1 + Cos[c + d*x]], x, 4, (Sqrt[2]*ArcSin[Tan[(c + d*x)/2]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[1 + Cos[c + d*x]]) + (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[1 + Cos[c + d*x]])}
{Sec[c + d*x]^(3/2)/Sqrt[1 + Cos[c + d*x]], x, 3, -((Sqrt[2]*ArcSin[Tan[(c + d*x)/2]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]])}
{Sec[c + d*x]^(1/2)/Sqrt[1 + Cos[c + d*x]], x, 2, (Sqrt[2]*ArcSin[Tan[(c + d*x)/2]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d}
{1/(Sqrt[1 + Cos[c + d*x]]*Sec[c + d*x]^(1/2)), x, 5, -((Sqrt[2]*ArcSin[Tan[(1/2)*(c + d*x)]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d) + (2*ArcTan[Sin[c + d*x]/(Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d}
{1/(Sqrt[1 + Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 5, (Sqrt[2]*ArcSin[Tan[(1/2)*(c + d*x)]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (ArcTan[Sin[c + d*x]/(Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + Sin[c + d*x]/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{Sec[c + d*x]^(7/2)/Sqrt[a + a*Cos[c + d*x]], x, 5, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (26*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^(5/2)/Sqrt[a + a*Cos[c + d*x]], x, 4, (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^(3/2)/Sqrt[a + a*Cos[c + d*x]], x, 3, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^(1/2)/Sqrt[a + a*Cos[c + d*x]], x, 2, (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)}
{1/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(1/2)), x, 5, (2*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)}
{1/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 5, -((ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + Sin[c + d*x]/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(3/2), x, 5, (11*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - (19*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (7*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(3/2), x, 4, -((7*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (5*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(3/2), x, 3, (3*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(1/2)), x, 3, (ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)), x, 5, (2*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) - (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)), x, 6, -((3*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d)) + (9*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (3*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(5/2), x, 6, (163*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - (299*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (17*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (95*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(5/2), x, 5, -((75*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (13*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (49*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(5/2), x, 4, (19*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - (9*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(1/2)), x, 4, (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + (5*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 4, (3*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + (3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)), x, 6, (2*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) - (43*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) - (11*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)), x, 7, -((5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d)) + (115*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) - (15*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (35*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(7/2), x, 7, (1015*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - (629*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (23*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - (109*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + (193*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(7/2), x, 6, -((363*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d)) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (19*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - (199*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + (691*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(7/2), x, 5, (63*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) - (5*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - (103*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(1/2)), x, 5, (13*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) + Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) + Sin[c + d*x]/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - (5*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)), x, 5, (7*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)) + (7*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + (7*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)), x, 5, (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) + Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)) + (5*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + (5*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)), x, 7, (2*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(7/2)*d) - (177*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)) - (17*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) - (49*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(9/2)), x, 8, -((7*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(7/2)*d)) + (637*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)) - (7*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) - (259*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (189*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(5/2)), x, 6, (45*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(1024*Sqrt[2]*a^(9/2)*d) - Sin[c + d*x]/(8*d*(a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(7/2)) + (3*Sin[c + d*x])/(32*a*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)) + (15*Sin[c + d*x])/(256*a^2*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + (45*Sin[c + d*x])/(1024*a^3*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(7/2)), x, 6, (35*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(1024*Sqrt[2]*a^(9/2)*d) + Sin[c + d*x]/(8*d*(a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(7/2)) + (7*Sin[c + d*x])/(96*a*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)) + (35*Sin[c + d*x])/(768*a^2*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + (35*Sin[c + d*x])/(1024*a^3*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Sec[c+d x]^(m/4) / (a+a Cos[c+d x])^(n/2)*)


{Sec[c + d*x]^(5/4)*(a + a*Cos[c + d*x])^(3/2), x, 2, (4*a^2*Sec[c + d*x]^(1/4)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}


(* ::Section:: *)
(*Cos[c+d x]^m (A+B Cos[c+d x]) (a+a Cos[c+d x])^n*)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]) (a+a Cos[c+d x])^n*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{Cos[c + d*x]^5*(A + B*Cos[c + d*x]), x, 6, (5*B*x)/16 + (8*A*Sin[c + d*x])/(15*d) + (5*B*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (4*A*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (5*B*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) + (B*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x]), x, 5, (3*A*x)/8 + (8*B*Sin[c + d*x])/(15*d) + (3*A*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (4*B*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x]), x, 4, (3*B*x)/8 + (2*A*Sin[c + d*x])/(3*d) + (3*B*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x]), x, 3, (A*x)/2 + (2*B*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x]), x, 2, (B*x)/2 + (A*Sin[c + d*x])/d + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x]), x, 2, A*x + (B*Sin[c + d*x])/d}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x]), x, 2, B*x + (A*ArcTanh[Sin[c + d*x]])/d}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x]), x, 3, (B*ArcTanh[Sin[c + d*x]])/d + (A*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x]), x, 4, (A*ArcTanh[Sin[c + d*x]])/(2*d) + (B*Tan[c + d*x])/d + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x]), x, 5, (B*ArcTanh[Sin[c + d*x]])/(2*d) + (2*A*Tan[c + d*x])/(3*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x]), x, 6, (3*A*ArcTanh[Sin[c + d*x]])/(8*d) + (2*B*Tan[c + d*x])/(3*d) + (3*A*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + B*Cos[c + d*x]), x, 7, (3*B*ArcTanh[Sin[c + d*x]])/(8*d) + (8*A*Tan[c + d*x])/(15*d) + (3*B*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (4*A*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (A*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}


{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 5, (3/8)*a*(A + B)*x + (2*a*(5*A + 4*B)*Sin[c + d*x])/(15*d) + (3*a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(5*A + 4*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a*(A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 4, (1/8)*a*(4*A + 3*B)*x + (2*a*(A + B)*Sin[c + d*x])/(3*d) + (a*(4*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(A + B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (a*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 3, (1/2)*a*(A + B)*x + (a*(3*A + 2*B)*Sin[c + d*x])/(3*d) + (a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 2, (1/2)*a*(2*A + B)*x + (a*(A + B)*Sin[c + d*x])/d + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 3, (a*A + a*B)*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*B*Sin[c + d*x])/d}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 3, a*B*x + (a*(A + B)*ArcTanh[Sin[c + d*x]])/d + (a*A*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 4, (a*(A + 2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(A + B)*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 5, (a*(A + B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A + 3*B)*Tan[c + d*x])/(3*d) + (a*(A + B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 6, (a*(3*A + 4*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*(A + B)*Tan[c + d*x])/(3*d) + (a*(3*A + 4*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(A + B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 6, (1/16)*a^2*(12*A + 11*B)*x + (2*a^2*(9*A + 8*B)*Sin[c + d*x])/(15*d) + (a^2*(12*A + 11*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(9*A + 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a^2*(12*A + 11*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*(6*A + 7*B)*Cos[c + d*x]^4*Sin[c + d*x])/(30*d) + (a*B*Cos[c + d*x]^4*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 5, (1/8)*a^2*(7*A + 6*B)*x + (2*a^2*(10*A + 9*B)*Sin[c + d*x])/(15*d) + (a^2*(7*A + 6*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(10*A + 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a^2*(5*A + 6*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (a*B*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 4, (1/8)*a^2*(8*A + 7*B)*x + (a^2*(8*A + 7*B)*Sin[c + d*x])/(6*d) + (a^2*(8*A + 7*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*A - B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (B*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 3, (1/2)*a^2*(3*A + 2*B)*x + (2*a^2*(3*A + 2*B)*Sin[c + d*x])/(3*d) + (a^2*(3*A + 2*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (B*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 4, (1/2)*a^2*(4*A + 3*B)*x + (a^2*A*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*A + 3*B)*Sin[c + d*x])/(2*d) + (a*B*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 4, (a^2*A + 2*a^2*B)*x + (a^2*(2*A + B)*ArcTanh[Sin[c + d*x]])/d - (a^2*(A - B)*Sin[c + d*x])/d + (a*A*(a + a*Cos[c + d*x])*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 4, a^2*B*x + (a^2*(3*A + 4*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(3*A + 2*B)*Tan[c + d*x])/(2*d) + (a*A*(a + a*Sec[c + d*x])*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 5, (a^2*(2*A + 3*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a^2*(2*A + 3*B)*Tan[c + d*x])/(3*d) + (a^2*(2*A + 3*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 6, (a^2*(7*A + 8*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(4*A + 5*B)*Tan[c + d*x])/(3*d) + (a^2*(7*A + 8*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(5*A + 4*B)*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (a*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 6, (1/16)*a^3*(26*A + 23*B)*x + (2*a^3*(19*A + 17*B)*Sin[c + d*x])/(15*d) + (a^3*(26*A + 23*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(19*A + 17*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a^3*(22*A + 21*B)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a^2*(3*A + 4*B)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (a*B*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 5, (1/8)*a^3*(15*A + 13*B)*x + (a^3*(15*A + 13*B)*Sin[c + d*x])/(6*d) + (a^3*(15*A + 13*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*(15*A + 13*B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5*A - B)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (B*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 4, (5/8)*a^3*(4*A + 3*B)*x + (5*a^3*(4*A + 3*B)*Sin[c + d*x])/(6*d) + (5*a^3*(4*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*(4*A + 3*B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (B*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 5, (1/2)*a^3*(7*A + 5*B)*x + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(A + B)*Sin[c + d*x])/(2*d) + (a^2*(3*A + 5*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (a*B*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 5, (1/2)*a^3*(6*A + 7*B)*x + (a^3*(3*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*B*Sin[c + d*x])/(2*d) - (a^2*(2*A - B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (a*A*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 5, (a^3*A + 3*a^3*B)*x + (a^3*(7*A + 6*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*A*Sin[c + d*x])/(2*d) + (a^2*(2*A + B)*(a + a*Cos[c + d*x])*Tan[c + d*x])/d + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 5, a^3*B*x + (a^3*(5*A + 7*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(A + B)*Tan[c + d*x])/(2*d) + (a^2*(5*A + 3*B)*(a + a*Sec[c + d*x])*Tan[c + d*x])/(6*d) + (a*A*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 6, (5*a^3*(3*A + 4*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(3*A + 4*B)*Tan[c + d*x])/(6*d) + (5*a^3*(3*A + 4*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (a*(3*A + 4*B)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (A*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 7, (a^3*(13*A + 15*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(38*A + 45*B)*Tan[c + d*x])/(15*d) + (a^3*(13*A + 15*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^3*(43*A + 45*B)*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + (a^2*(7*A + 5*B)*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}


{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 7, (1/16)*a^4*(49*A + 44*B)*x + (2*a^4*(252*A + 227*B)*Sin[c + d*x])/(105*d) + (a^4*(49*A + 44*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^4*(252*A + 227*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (a^4*(301*A + 276*B)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (7*a^3*(A + B)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (a^2*(7*A + 10*B)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(42*d) + (a*B*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 6, (7/16)*a^4*(8*A + 7*B)*x + (7*a^4*(8*A + 7*B)*Sin[c + d*x])/(12*d) + (7*a^4*(8*A + 7*B)*Cos[c + d*x]*Sin[c + d*x])/(48*d) + (7*a^2*(8*A + 7*B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(120*d) + (a*(8*A + 7*B)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(40*d) + ((6*A - B)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (B*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 5, (7/8)*a^4*(5*A + 4*B)*x + (7*a^4*(5*A + 4*B)*Sin[c + d*x])/(6*d) + (7*a^4*(5*A + 4*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (7*a^2*(5*A + 4*B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (a*(5*A + 4*B)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (B*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 6, (1/8)*a^4*(48*A + 35*B)*x + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(8*A + 7*B)*Sin[c + d*x])/(8*d) + (a^3*(32*A + 35*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(24*d) + (a^2*(4*A + 7*B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (a*B*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 6, (1/2)*a^4*(13*A + 12*B)*x + (a^4*(4*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(A + 2*B)*Sin[c + d*x])/(2*d) - (a^3*(3*A - 8*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d) - (a^2*(3*A - B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (a*A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 6, (1/2)*a^4*(8*A + 13*B)*x + (a^4*(13*A + 8*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(A - B)*Sin[c + d*x])/(2*d) - (a^3*(6*A + B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (a^2*(5*A + 2*B)*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 6, (a^4*A + 4*a^4*B)*x + (a^4*(12*A + 13*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(2*A + B)*Sin[c + d*x])/(2*d) + (a^3*(11*A + 9*B)*(a + a*Cos[c + d*x])*Tan[c + d*x])/(3*d) + (a^2*(2*A + B)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 6, a^4*B*x + (a^4*(35*A + 48*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^4*(7*A + 8*B)*Tan[c + d*x])/(8*d) + (a^3*(35*A + 32*B)*(a + a*Sec[c + d*x])*Tan[c + d*x])/(24*d) + (a^2*(7*A + 4*B)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (a*A*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 7, (7*a^4*(4*A + 5*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (7*a^4*(4*A + 5*B)*Tan[c + d*x])/(6*d) + (7*a^4*(4*A + 5*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (7*a^2*(4*A + 5*B)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + (a*(4*A + 5*B)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (A*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)}
{Sec[c + d*x]^7*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 8, (7*a^4*(7*A + 8*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^4*(72*A + 83*B)*Tan[c + d*x])/(15*d) + (7*a^4*(7*A + 8*B)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^4*(159*A + 176*B)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + (a^3*(73*A + 72*B)*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (a^2*(3*A + 2*B)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)}


{(-B*(1/(1 + 1)) + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 2, (a*B*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{(-B*(4/(4 + 1)) + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 1, (B*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)}
{(-B*(n/(n + 1)) + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^n, x, 1, (B*(a + a*Cos[c + d*x])^n*Sin[c + d*x])/(d*(1 + n))}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 5, -((3*(4*A - 5*B)*x)/(8*a)) + (8*(A - B)*Sin[c + d*x])/(3*a*d) - (3*(4*A - 5*B)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (4*(A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d) - ((4*A - 5*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 4, (3*(A - B)*x)/(2*a) - (2*(3*A - 4*B)*Sin[c + d*x])/(3*a*d) + (3*(A - B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((3*A - 4*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 3, -(((2*A - 3*B)*x)/(2*a)) + (2*(A - B)*Sin[c + d*x])/(a*d) - ((2*A - 3*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 3, ((A - B)*x)/a + (B*Sin[c + d*x])/(a*d) - ((A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 2, (B*x)/a + ((A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 2, (A*ArcTanh[Sin[c + d*x]])/(a*d) - ((A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 4, -(((A - B)*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*A - B)*Tan[c + d*x])/(a*d) - ((A - B)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 5, ((3*A - 2*B)*ArcTanh[Sin[c + d*x]])/(2*a*d) - (2*(A - B)*Tan[c + d*x])/(a*d) + ((3*A - 2*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 6, -((3*(A - B)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + (2*(4*A - 3*B)*Tan[c + d*x])/(3*a*d) - (3*(A - B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((4*A - 3*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d) - ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 5, ((7*A - 10*B)*x)/(2*a^2) - (8*(2*A - 3*B)*Sin[c + d*x])/(3*a^2*d) + ((7*A - 10*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (4*(2*A - 3*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((7*A - 10*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 4, -(((4*A - 7*B)*x)/(2*a^2)) + (2*(5*A - 8*B)*Sin[c + d*x])/(3*a^2*d) - ((4*A - 7*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((5*A - 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 4, ((A - 2*B)*x)/a^2 - ((A - 4*B)*Sin[c + d*x])/(3*a^2*d) + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((A - 2*B)*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 3, (B*x)/a^2 + ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((A - 4*B)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 2, ((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((A + 2*B)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 3, (A*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((4*A - B)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 5, -(((2*A - B)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + (2*(5*A - 2*B)*Tan[c + d*x])/(3*a^2*d) - ((A - B)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((2*A - B)*Tan[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 6, ((7*A - 4*B)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*(8*A - 5*B)*Tan[c + d*x])/(3*a^2*d) + ((7*A - 4*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((8*A - 5*B)*Sec[c + d*x]*Tan[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 7, -(((10*A - 7*B)*ArcTanh[Sin[c + d*x]])/(2*a^2*d)) + (8*(3*A - 2*B)*Tan[c + d*x])/(3*a^2*d) - ((10*A - 7*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + (4*(3*A - 2*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d) - ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((10*A - 7*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^5*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 6, ((13*A - 23*B)*x)/(2*a^3) - (8*(19*A - 34*B)*Sin[c + d*x])/(15*a^3*d) + ((13*A - 23*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (4*(19*A - 34*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^3*d) + ((A - B)*Cos[c + d*x]^5*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((8*A - 13*B)*Cos[c + d*x]^4*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((13*A - 23*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 5, -(((6*A - 13*B)*x)/(2*a^3)) + (8*(9*A - 19*B)*Sin[c + d*x])/(15*a^3*d) - ((6*A - 13*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((6*A - 11*B)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (4*(9*A - 19*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 5, ((A - 3*B)*x)/a^3 - ((7*A - 27*B)*Sin[c + d*x])/(15*a^3*d) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*A - 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - 3*B)*Sin[c + d*x])/(a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 4, (B*x)/a^3 + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*A - 7*B)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (2*(A - 11*B)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 3, ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (B*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) + ((3*A + 7*B)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 3, ((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*A + 3*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*A + 3*B)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 4, (A*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*A - 2*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (2*(11*A - B)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 6, -(((3*A - B)*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (2*(36*A - 11*B)*Tan[c + d*x])/(15*a^3*d) - ((A - B)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*A - 4*B)*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((3*A - B)*Tan[c + d*x])/(a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 7, ((13*A - 6*B)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (8*(19*A - 9*B)*Tan[c + d*x])/(15*a^3*d) + ((13*A - 6*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*A - 6*B)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (4*(19*A - 9*B)*Sec[c + d*x]*Tan[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^5*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 6, -(((8*A - 21*B)*x)/(2*a^4)) + (8*(83*A - 216*B)*Sin[c + d*x])/(105*a^4*d) - ((8*A - 21*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) + ((A - B)*Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((A - 2*B)*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3) + ((52*A - 129*B)*Cos[c + d*x]^3*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + (4*(83*A - 216*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 6, ((A - 4*B)*x)/a^4 - ((55*A - 244*B)*Sin[c + d*x])/(105*a^4*d) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((5*A - 12*B)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((25*A - 88*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - ((A - 4*B)*Sin[c + d*x])/(a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 5, (B*x)/a^4 + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A - 10*B)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((6*A - 55*B)*Cos[c + d*x]*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + (2*(3*A - 80*B)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 4, -(((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4)) + ((4*A + 3*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - (8*(4*A + 3*B)*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + ((4*A + 3*B)*Sin[c + d*x])/(15*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 4, ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((A + 6*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((8*A + 13*B)*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + ((8*A + 13*B)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 4, ((A - B)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A + 4*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (2*(3*A + 4*B)*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + (2*(3*A + 4*B)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 5, (A*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((A - B)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((10*A - 3*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - ((55*A - 6*B)*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - (2*(80*A - 3*B)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 7, -(((4*A - B)*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (8*(83*A - 20*B)*Tan[c + d*x])/(105*a^4*d) - ((A - B)*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((12*A - 5*B)*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - ((88*A - 25*B)*Tan[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - ((4*A - B)*Tan[c + d*x])/(a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 8, ((21*A - 8*B)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (8*(216*A - 83*B)*Tan[c + d*x])/(105*a^4*d) + ((21*A - 8*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((2*A - B)*Sec[c + d*x]*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3) - ((129*A - 52*B)*Sec[c + d*x]*Tan[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - (4*(216*A - 83*B)*Sec[c + d*x]*Tan[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}


{(-B*(-3/(-3 + 1)) + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 1, -((B*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^3))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]) (a+a Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 5, (4*a*(9*A + 8*B)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(9*A + 8*B)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*B*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (8*(9*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (4*(9*A + 8*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 4, (2*a*(7*A + 6*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*B*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) - (4*(7*A + 6*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A + 6*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*a*(5*A + 7*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*A - 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*a*(3*A + B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*Sqrt[2]*a*A*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 2, (Sqrt[2]*a*(A + 2*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 3, (a*(3*A + 4*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(3*A + 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 4, (a*(5*A + 6*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(5*A + 6*B)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(5*A + 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 6, (4*a^2*(187*A + 168*B)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(187*A + 168*B)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(11*A + 12*B)*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (8*a*(187*A + 168*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a*B*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (4*(187*A + 168*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 5, (2*a^2*(39*A + 34*B)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(9*A + 10*B)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(39*A + 34*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*B*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d) + (2*(39*A + 34*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 4, (8*a^2*(21*A + 19*B)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(21*A + 19*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A - 2*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*B*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 3, (8*a^2*(5*A + 3*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(5*A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 3, (2*Sqrt[2]*a^2*A*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(3*A + 4*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 3, (Sqrt[2]*a^2*(3*A + 2*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(A - 2*B)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 3, (a^2*(7*A + 12*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(5*A + 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 4, (a^2*(11*A + 14*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(11*A + 14*B)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(7*A + 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 5, (a^2*(75*A + 88*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(32*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(75*A + 88*B)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(75*A + 88*B)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(9*A + 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 6, (2*a^3*(803*A + 710*B)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(209*A + 194*B)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a^2*(803*A + 710*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a^2*(11*A + 14*B)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d) + (2*a*(803*A + 710*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*a*B*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (64*a^3*(15*A + 13*B)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(15*A + 13*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(15*A + 13*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (2*(9*A - 2*B)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*B*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (64*a^3*(7*A + 5*B)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(7*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(7*A + 5*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*B*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (2*Sqrt[2]*a^3*A*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(35*A + 32*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(5*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (Sqrt[2]*a^3*(5*A + 2*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(3*A + 14*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(3*A - 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (a^3*(19*A + 20*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - (a^3*(9*A - 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(7*A + 4*B)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (a^3*(25*A + 38*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(49*A + 54*B)*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(3*A + 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (a^3*(163*A + 200*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(32*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(163*A + 200*B)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(95*A + 104*B)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(11*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 6, (a^3*(283*A + 326*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(64*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(283*A + 326*B)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(283*A + 326*B)*Sec[c + d*x]*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(157*A + 170*B)*Sec[c + d*x]^2*Tan[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(13*A + 10*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}


{(-B*(3/2/(3/2 + 1)) + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 1, (2*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 5, -((2*(A - B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + (4*(49*A - 37*B)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(7*A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(7*A - 31*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 4, (2*(A - B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) - (4*(5*A - 7*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*A - B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 3, -((2*(A - B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + (2*(3*A - 2*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 2, (2*(A - B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 3, -((2*(A - B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + (2*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 4, (2*(A - B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) - (Sqrt[2]*(A - 2*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) + (A*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 5, -((2*(A - B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + ((7*A - 4*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 6, -(((15*A - 19*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((651*A - 799*B)*Sin[c + d*x])/(105*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((63*A - 67*B)*Cos[c + d*x]^2*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((7*A - 11*B)*Cos[c + d*x]^3*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((273*A - 397*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(210*a^2*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 5, ((11*A - 15*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((65*A - 93*B)*Sin[c + d*x])/(15*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((5*A - 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((35*A - 39*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(30*a^2*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 4, -(((7*A - 11*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])) + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 13*B)*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((3*A - 7*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 3, ((3*A - 7*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((A - 5*B)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 2, ((A + 3*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 4, -(((5*A - B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])) + (2*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 5, ((9*A - 5*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - (Sqrt[2]*(3*A - 2*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A - B)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 6, -(((13*A - 9*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])) + ((19*A - 12*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*Sqrt[2]*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((7*A - 6*B)*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((2*A - B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 6, ((163*A - 283*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((13*A - 21*B)*Cos[c + d*x]^3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((985*A - 1729*B)*Sin[c + d*x])/(120*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((85*A - 157*B)*Cos[c + d*x]^2*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((475*A - 787*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*a^3*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 5, -(((75*A - 163*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((9*A - 17*B)*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((93*A - 197*B)*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 95*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 4, ((19*A - 75*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A - 13*B)*Cos[c + d*x]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((9*A - 49*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 3, ((5*A + 19*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((A - 9*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 3, ((3*A + 5*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A + 5*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 5, -(((43*A - 3*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])) + (2*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((11*A - 3*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 6, ((115*A - 43*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - (Sqrt[2]*(5*A - 2*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((15*A - 7*B)*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((35*A - 11*B)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 7, -(((219*A - 115*B)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])) + ((39*A - 20*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*Sqrt[2]*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - (7*(9*A - 5*B)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((19*A - 11*B)*Sec[c + d*x]*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((31*A - 15*B)*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}


{(-B*(-1/2/(-1/2 + 1)) + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 2, (2*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{(-B*(-5/2/(-5/2 + 1)) + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 1, -((2*B*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^(5/2)))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]) (a+a Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{(A + B*Cos[c + d*x])*Cos[c + d*x]^(5/2), x, 6, (6*A*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (10*B*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{(A + B*Cos[c + d*x])*Cos[c + d*x]^(3/2), x, 5, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*A*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{(A + B*Cos[c + d*x])*Cos[c + d*x]^(1/2), x, 4, (2*A*EllipticE[(1/2)*(c + d*x), 2])/d + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(A + B*Cos[c + d*x])/Cos[c + d*x]^(1/2), x, 3, (2*B*EllipticE[(1/2)*(c + d*x), 2])/d + (2*A*EllipticF[(1/2)*(c + d*x), 2])/d}
{(A + B*Cos[c + d*x])/Cos[c + d*x]^(3/2), x, 4, -((2*A*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/d + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(A + B*Cos[c + d*x])/Cos[c + d*x]^(5/2), x, 5, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*A*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(A + B*Cos[c + d*x])/Cos[c + d*x]^(7/2), x, 6, -((6*A*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*A*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 7, (2*a*(9*A + 7*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*a*(A + B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*a*(A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(A + B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*B*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 6, (6*a*(A + B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(7*A + 5*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(7*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(A + B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 5, (2*a*(5*A + 3*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(A + B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 4, (2*a*(A + B)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*(3*A + B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 4, -((2*a*(A - B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*(A + B)*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 5, -((2*a*(A + B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*(A + 3*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 6, -((2*a*(3*A + 5*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(A + B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])}


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 7, (4*a^2*(9*A + 8*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (4*a^2*(6*A + 5*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^2*(6*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a^2*(9*A + 8*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(9*A + 11*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*a*B*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 6, (4*a^2*(4*A + 3*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^2*(7*A + 6*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^2*(7*A + 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(7*A + 9*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*a*B*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 5, (4*a^2*(5*A + 4*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^2*(2*A + B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(5*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*B*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 5, (4*a^2*B*EllipticE[(1/2)*(c + d*x), 2])/d + (4*a^2*(3*A + 2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*a^2*(3*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 5, -((4*a^2*A*EllipticE[(1/2)*(c + d*x), 2])/d) + (4*a^2*(2*A + 3*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(5*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 6, -((4*a^2*(4*A + 5*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^2*(A + 2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(7*A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (4*a^2*(4*A + 5*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 7, -((4*a^2*(3*A + 4*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^2*(6*A + 7*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^2*(9*A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (4*a^2*(6*A + 7*B)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^2*(3*A + 4*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 8, (4*a^3*(17*A + 15*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (4*a^3*(121*A + 105*B)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (4*a^3*(121*A + 105*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(17*A + 15*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (20*a^3*(22*A + 21*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*a^2*(11*A + 15*B)*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(99*d) + (2*a*B*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 7, (4*a^3*(21*A + 17*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (4*a^3*(13*A + 11*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^3*(13*A + 11*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a^3*(24*A + 23*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a^2*(9*A + 13*B)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(63*d) + (2*a*B*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 6, (4*a^3*(9*A + 7*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^3*(21*A + 13*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^3*(42*A + 41*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a^2*(7*A + 11*B)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d) + (2*a*B*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 6, (4*a^3*(5*A + 9*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^3*(5*A + 3*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (4*a^3*(5*A - 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*a^2*(5*A - B)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 6, -((4*a^3*(A - B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (20*a^3*(A + B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (4*a^3*(4*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(7*A + 3*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 6, -((4*a^3*(9*A + 5*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^3*(3*A + 5*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (4*a^3*(21*A + 20*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(9*A + 5*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 7, -((4*a^3*(7*A + 9*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^3*(13*A + 21*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^3*(41*A + 42*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(7*A + 9*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(11*A + 7*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 8, -((4*a^3*(17*A + 21*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (4*a^3*(11*A + 13*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^3*(23*A + 24*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^3*(11*A + 13*B)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^3*(17*A + 21*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(13*A + 9*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 6, -((3*(5*A - 7*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*a*d)) + (5*(A - B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (5*(A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((5*A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 5, (3*(A - B)*EllipticE[(1/2)*(c + d*x), 2])/(a*d) - ((3*A - 5*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) - ((3*A - 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 4, -(((A - 3*B)*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) + ((A - B)*EllipticF[(1/2)*(c + d*x), 2])/(a*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 4, ((A - B)*EllipticE[(1/2)*(c + d*x), 2])/(a*d) + ((A + B)*EllipticF[(1/2)*(c + d*x), 2])/(a*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 5, -(((3*A - B)*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - ((A - B)*EllipticF[(1/2)*(c + d*x), 2])/(a*d) + ((3*A - B)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 6, (3*(A - B)*EllipticE[(1/2)*(c + d*x), 2])/(a*d) + ((5*A - 3*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + ((5*A - 3*B)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (3*(A - B)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 7, -((7*(5*A - 8*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*a^2*d)) + (5*(2*A - 3*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (5*(2*A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - (7*(5*A - 8*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) + ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((2*A - 3*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 6, ((4*A - 7*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) - (5*(A - 2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (5*(A - 2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((4*A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 5, -(((A - 4*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + ((2*A - 5*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((2*A - 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 5, -((B*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + ((A + 2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 5, (A*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + ((2*A + B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 6, -(((4*A - B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) - ((5*A - 2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + ((4*A - B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((5*A - 2*B)*Sin[c + d*x])/(3*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 7, ((7*A - 4*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (5*(2*A - B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (5*(2*A - B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((7*A - 4*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((7*A - 4*B)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^(9/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 8, -((7*(17*A - 33*B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + ((11*A - 21*B)*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) + ((11*A - 21*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - (7*(17*A - 33*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) + ((A - B)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((7*A - 12*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (3*(11*A - 21*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 7, (7*(7*A - 17*B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) - ((13*A - 33*B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((13*A - 33*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A - 2*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) + (7*(7*A - 17*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 6, -(((9*A - 49*B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + ((3*A - 13*B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((3*A - 8*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((3*A - 13*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 6, -(((A + 9*B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + ((A + 3*B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A - 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A + 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 6, ((A - B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + ((A + B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A + 4*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 6, ((9*A + B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + ((3*A + B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((6*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((9*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 7, -(((49*A - 9*B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) - ((13*A - 3*B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + ((49*A - 9*B)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B)*Sin[c + d*x])/(6*a^2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 8, (7*(17*A - 7*B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + ((33*A - 13*B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + ((33*A - 13*B)*Sin[c + d*x])/(6*a^3*d*Cos[c + d*x]^(3/2)) - (7*(17*A - 7*B)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - (7*(17*A - 7*B)*Sin[c + d*x])/(30*a^2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]) (a+a Cos[c+d x])^(n/2)*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 5, (5*Sqrt[a]*(8*A + 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*d) + (5*a*(8*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (5*a*(8*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(8*A + 7*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 4, (Sqrt[a]*(6*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) + (a*(6*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(6*A + 5*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 3, (Sqrt[a]*(4*A + 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) + (a*(4*A + 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 2, (Sqrt[a]*(2*A + B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (a*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*Sqrt[a]*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(2*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(4*A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(4*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 4, (2*a*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(6*A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(6*A + 7*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(6*A + 7*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 5, (a^(3/2)*(88*A + 75*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*d) + (a^2*(88*A + 75*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(88*A + 75*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(8*A + 9*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 4, (a^(3/2)*(14*A + 11*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) + (a^2*(14*A + 11*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(6*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 3, (a^(3/2)*(12*A + 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) + (a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 3, (a^(3/2)*(2*A + 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d - (a^2*(2*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 3, (2*a^(3/2)*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a^2*(4*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 3, (8*a^2*(3*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 4, (2*a^2*(8*A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 63*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(52*A + 63*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 5, (2*a^2*(10*A + 9*B)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(34*A + 39*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(34*A + 39*B)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(34*A + 39*B)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 6, (a^(5/2)*(326*A + 283*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(128*d) + (a^3*(326*A + 283*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(326*A + 283*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(170*A + 157*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(10*A + 13*B)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (a*B*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (a^(5/2)*(200*A + 163*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*d) + (a^3*(200*A + 163*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(104*A + 95*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(8*A + 11*B)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d) + (a*B*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (a^(5/2)*(38*A + 25*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) + (a^3*(54*A + 49*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(2*A + 3*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (a^(5/2)*(20*A + 19*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) - (a^3*(4*A - 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(4*A - B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (a^(5/2)*(2*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d - (a^3*(14*A + 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(2*A + B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (2*a^(5/2)*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a^3*(32*A + 35*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (64*a^3*(5*A + 7*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(5*A + 7*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 7*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (2*a^3*(124*A + 135*B)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(292*A + 345*B)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(292*A + 345*B)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(4*A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}
{Cos[c + d*x]^(-13/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 6, (2*a^3*(194*A + 209*B)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(710*A + 803*B)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(710*A + 803*B)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(710*A + 803*B)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(14*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 5, -(((4*A - 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*Sqrt[a]*d)) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((4*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 4, ((2*A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 3, (2*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 2, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 3, (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 4, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(13*A - 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 5, ((2*A - 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(3/2)*d) - ((5*A - 9*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 4, (2*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(3/2)*d) + ((A - 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 2, ((3*A + B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 3, -(((7*A - 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 4, ((11*A - 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B)*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((19*A - 15*B)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 6, ((2*A - 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(5/2)*d) - ((43*A - 115*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((7*A - 15*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((11*A - 35*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 5, (2*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(5/2)*d) + ((3*A - 43*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A - 11*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 3, ((5*A + 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A + 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 3, ((19*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((9*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 4, -(((75*A - 19*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 5, ((163*A - 75*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((95*A - 39*B)*Sin[c + d*x])/(48*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((299*A - 147*B)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 7, ((2*A - 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(7/2)*d) - ((177*A - 637*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((3*A - 7*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) + ((79*A - 259*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) - (7*(7*A - 27*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 6, (2*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(7/2)*d) + ((5*A - 177*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((5*A - 17*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A - 49*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 4, ((7*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((7*A + 5*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) + ((7*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 4, ((13*A + 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((A + 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((5*A - 17*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 4, ((63*A + 13*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - ((5*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((103*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 5, -((3*(121*A - 21*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d)) - ((A - B)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)) - ((19*A - 7*B)*Sin[c + d*x])/(48*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((199*A - 43*B)*Sin[c + d*x])/(192*a^2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((691*A - 103*B)*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 6, ((1015*A - 363*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sin[c + d*x])/(6*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)) - ((23*A - 11*B)*Sin[c + d*x])/(48*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((109*A - 41*B)*Sin[c + d*x])/(64*a^2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((579*A - 199*B)*Sin[c + d*x])/(192*a^3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((1887*A - 691*B)*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]) (a+a Cos[c+d x])^(n/3)*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(2/3), x, 2, (1/10)*a*(5*A + 2*B)*Int[1/(a + a*Cos[c + d*x])^(1/3), x] + (3*a*(5*A + 2*B)*Sin[c + d*x])/(10*d*(a + a*Cos[c + d*x])^(1/3)) + (3*B*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*d)}
{(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/3), x, 2, (-(1/4))*a*(4*A + B)*Int[1/(a + a*sin[c + Pi/2 + d*x])^(2/3), x] + (3*a*(4*A + B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(2/3)) + (3*B*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/3), x, 1, (1/2)*(2*A - B)*Int[1/(a + a*Cos[c + d*x])^(1/3), x] + (3*B*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(1/3))}
{(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(2/3), x, 1, (A - 2*B)*Int[1/(a + a*sin[c + Pi/2 + d*x])^(2/3), x] + (3*B*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^(2/3))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Sec[c+d x]^(m/2) (A+B Cos[c+d x]) (a+a Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 7, -((6*A*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 6, -((2*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 5, -((2*A*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2), x, 4, (2*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d}
{(A + B*Cos[c + d*x])/Sec[c + d*x]^(1/2), x, 5, (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/Sec[c + d*x]^(3/2), x, 6, (6*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/Sec[c + d*x]^(5/2), x, 7, (6*A*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*B*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (10*B*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 8, (-6*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (6*a*(A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*A + 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*(A + B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 7, (-2*a*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(A + B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 6, (-2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 5, (-2*a*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 5, (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 6, (2*a*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2), x, 7, (6*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*B*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2), x, 9, (-4*a^2*(8*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(8*A + 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^2*(5*A + 6*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (4*a^2*(8*A + 9*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (2*a^2*(11*A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*a*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 8, (-4*a^2*(3*A + 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(6*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(3*A + 4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^2*(6*A + 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(9*A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 7, (-4*a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^2*(4*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(7*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 6, (-4*a^2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(2*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(5*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 6, (4*a^2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(3*A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(3*A - B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 6, (4*a^2*(5*A + 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(2*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(5*A + 7*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*B*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 7, (4*a^2*(4*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(7*A + 6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(7*A + 9*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*a*B*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(3/2)) + (4*a^2*(7*A + 6*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2), x, 8, (4*a^2*(9*A + 8*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(6*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(9*A + 11*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*a*B*(a + a*Cos[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^(5/2)) + (4*a^2*(9*A + 8*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (4*a^2*(6*A + 5*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(13/2), x, 10, (-4*a^3*(15*A + 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(105*A + 121*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(15*A + 17*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(105*A + 121*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (4*a^3*(15*A + 17*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (20*a^3*(21*A + 22*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d) + (2*a^2*(15*A + 11*B)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2), x, 9, (-4*a^3*(17*A + 21*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(17*A + 21*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(11*A + 13*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (4*a^3*(23*A + 24*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*a^2*(13*A + 9*B)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 8, (-4*a^3*(7*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 21*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(7*A + 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(41*A + 42*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a^2*(11*A + 7*B)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 7, (-4*a^3*(9*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(21*A + 20*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a^2*(9*A + 5*B)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 7, (-4*a^3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (20*a^3*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(4*A + B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(7*A + 3*B)*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 7, (4*a^3*(5*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(5*A - 6*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*a^2*(5*A - B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 7, (4*a^3*(9*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(21*A + 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(42*A + 41*B)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(7*A + 11*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]) + (2*a*B*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 8, (4*a^3*(21*A + 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(13*A + 11*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(24*A + 23*B)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (2*a^2*(9*A + 13*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(3/2)) + (2*a*B*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (4*a^3*(13*A + 11*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2), x, 9, (4*a^3*(17*A + 15*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(121*A + 105*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (20*a^3*(22*A + 21*B)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*a^2*(11*A + 15*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(5/2)) + (2*a*B*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (4*a^3*(17*A + 15*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (4*a^3*(121*A + 105*B)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x]), x, 8, (-3*(7*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - (5*(A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(7*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) - (5*(A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((7*A - 5*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x]), x, 7, (3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((5*A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - (3*(A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((5*A - 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x]), x, 6, -(((3*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) - ((A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x]), x, 5, ((A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]), x, 5, -(((A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 6, (3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((3*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) - ((3*A - 5*B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)), x, 7, (-3*(5*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) - ((5*A - 7*B)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(A - B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])}


{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^2, x, 8, ((7*A - 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (5*(2*A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((7*A - 4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (5*(2*A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((7*A - 4*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^2, x, 7, -(((4*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) - ((5*A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((4*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((5*A - 2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^2, x, 6, (A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((2*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - (A*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]), x, 6, -((B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) + (B*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)), x, 6, -(((A - 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((2*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((2*A - 5*B)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)), x, 7, ((4*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - (5*(A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) + ((4*A - 7*B)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) - (5*(A - 2*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])}


{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^3, x, 9, (7*(17*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((33*A - 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - (7*(17*A - 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((33*A - 13*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a^3*d) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - (7*(17*A - 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x]))}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^3, x, 8, -((49*A - 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((49*A - 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x]))}
{((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^3, x, 7, ((9*A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]) - ((6*A - B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((9*A + B)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]), x, 7, ((A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]) + ((A + 4*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)), x, 7, -((A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)) + ((A - 6*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) + ((A + 9*B)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)), x, 7, -((9*A - 49*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A - 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)) + ((3*A - 8*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((3*A - 13*B)*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)), x, 8, (7*(7*A - 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 33*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)) + ((A - 2*B)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) + (7*(7*A - 17*B)*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) - ((13*A - 33*B)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)), x, 9, -((7*(17*A - 33*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) + ((11*A - 21*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) + ((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)) + ((7*A - 12*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) + (3*(11*A - 21*B)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) - (7*(17*A - 33*B)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) + ((11*A - 21*B)*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Sec[c+d x]^(m/2) (A+B Cos[c+d x]) (a+a Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2), x, 5, (32*a*(8*A + 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(8*A + 9*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(8*A + 9*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(8*A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 4, (16*a*(6*A + 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(6*A + 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(6*A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 3, (4*a*(4*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(4*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 2, (2*a*(2*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 3, (2*Sqrt[a]*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 3, (Sqrt[a]*(2*A + B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 4, (Sqrt[a]*(4*A + 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*B*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(4*A + 3*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2), x, 5, (Sqrt[a]*(6*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*B*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a*(6*A + 5*B)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(6*A + 5*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(13/2), x, 6, (32*a^2*(168*A + 187*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(168*A + 187*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(168*A + 187*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(168*A + 187*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(12*A + 11*B)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2), x, 5, (16*a^2*(34*A + 39*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(34*A + 39*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(34*A + 39*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(10*A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 4, (4*a^2*(52*A + 63*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 63*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 3, (8*a^2*(3*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 4, (2*a^(3/2)*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(4*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 4, (a^(3/2)*(2*A + 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^2*(2*A - B)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 4, (a^(3/2)*(12*A + 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*(4*A + 5*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 5, (a^(3/2)*(14*A + 11*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(6*A + 7*B)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (a^2*(14*A + 11*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2), x, 6, (a^(3/2)*(88*A + 75*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(8*A + 9*B)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (a^2*(88*A + 75*B)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(88*A + 75*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(15/2), x, 7, (32*a^3*(4184*A + 4615*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(4184*A + 4615*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(4184*A + 4615*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(4184*A + 4615*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(280*A + 299*B)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(1287*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(16*A + 13*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(143*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(13/2)*Sin[c + d*x])/(13*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(13/2), x, 6, (16*a^3*(710*A + 803*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(710*A + 803*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(710*A + 803*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(194*A + 209*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(14*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2), x, 5, (4*a^3*(292*A + 345*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(292*A + 345*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(124*A + 135*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(4*A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 4, (64*a^3*(5*A + 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(5*A + 7*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*(5*A + 7*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 5, (2*a^(5/2)*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(32*A + 35*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 5, (a^(5/2)*(2*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^3*(14*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*(2*A + B)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 5, (a^(5/2)*(20*A + 19*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^3*(4*A - 9*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(4*A - B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 5, (a^(5/2)*(38*A + 25*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^3*(54*A + 49*B)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*(2*A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 6, (a^(5/2)*(200*A + 163*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(104*A + 95*B)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(8*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sec[c + d*x]^(3/2)) + (a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)) + (a^3*(200*A + 163*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2), x, 7, (a^(5/2)*(326*A + 283*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(170*A + 157*B)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*(10*A + 13*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d*Sec[c + d*x]^(5/2)) + (a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(5/2)) + (a^3*(326*A + 283*B)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^3*(326*A + 283*B)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{((A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2))/Sqrt[a + a*Cos[c + d*x]], x, 7, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(257*A - 129*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(29*A - 93*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(19*A - 3*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2))/Sqrt[a + a*Cos[c + d*x]], x, 6, (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A - 91*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A - 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/Sqrt[a + a*Cos[c + d*x]], x, 5, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/Sqrt[a + a*Cos[c + d*x]], x, 4, (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/Sqrt[a + a*Cos[c + d*x]], x, 3, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2))/Sqrt[a + a*Cos[c + d*x]], x, 4, (2*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)}
{(A + B*Cos[c + d*x])/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(1/2)), x, 5, ((2*A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 6, -((4*A - 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (B*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((4*A - B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}

{((a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]], x, 5, ((2*A*b + 2*a*B - b*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(a*A - A*b - a*B + b*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (b*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{((A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2), x, 7, ((19*A - 15*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((1201*A - 1029*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((397*A - 273*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((67*A - 63*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((11*A - 7*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(3/2), x, 6, -((15*A - 11*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((147*A - 95*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 35*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 5*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(3/2), x, 5, ((11*A - 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((19*A - 15*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(3/2), x, 4, -((7*A - 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2))/(a + a*Cos[c + d*x])^(3/2), x, 3, ((3*A + B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(1/2)), x, 5, (2*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A - 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)), x, 6, ((2*A - 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) - ((5*A - 9*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) - ((A - 3*B)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(5/2), x, 7, -((283*A - 163*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((2671*A - 1495*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((787*A - 475*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((21*A - 13*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((157*A - 85*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(5/2), x, 6, ((163*A - 75*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((299*A - 147*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((95*A - 39*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(5/2), x, 5, -((75*A - 19*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2))/(a + a*Cos[c + d*x])^(5/2), x, 4, ((19*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((9*A - B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(1/2)), x, 4, ((5*A + 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((5*A + 3*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 6, (2*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A - 43*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((3*A - 11*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)), x, 7, ((2*A - 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) - ((43*A - 115*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((7*A - 15*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) - ((11*A - 35*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(7/2), x, 7, ((1015*A - 363*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - ((1887*A - 691*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - ((23*A - 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((109*A - 41*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + ((579*A - 199*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(7/2), x, 6, -((3*(121*A - 21*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d)) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - ((19*A - 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((199*A - 43*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + ((691*A - 103*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2))/(a + a*Cos[c + d*x])^(7/2), x, 5, ((63*A + 13*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) - ((5*A - B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((103*A + 5*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(1/2)), x, 5, ((13*A + 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) + ((A + 3*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((5*A - 17*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)), x, 5, ((7*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)) + ((7*A + 5*B)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((7*A + 5*B)*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)), x, 7, (2*B*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(7/2)*d) + ((5*A - 177*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)) + ((5*A - 17*B)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((5*A - 49*B)*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)), x, 8, ((2*A - 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(7/2)*d) - ((177*A - 637*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)) + ((3*A - 7*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((79*A - 259*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) - (7*(7*A - 27*B)*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


(* ::Section:: *)
(*Cos[c+d x]^m (A+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{Cos[c + d*x]^6*(A + C*Cos[c + d*x]^2), x, 4, (5/128)*(8*A + 7*C)*x + (5*(8*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*(8*A + 7*C)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((8*A + 7*C)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (C*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)}
{Cos[c + d*x]^5*(A + C*Cos[c + d*x]^2), x, 3, ((7*A + 6*C)*Sin[c + d*x])/(7*d) + (C*Cos[c + d*x]^6*Sin[c + d*x])/(7*d) - (2*(7*A + 6*C)*Sin[c + d*x]^3)/(21*d) + ((7*A + 6*C)*Sin[c + d*x]^5)/(35*d)}
{Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2), x, 3, (1/16)*(6*A + 5*C)*x + ((6*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((6*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (C*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2), x, 3, ((5*A + 4*C)*Sin[c + d*x])/(5*d) + (C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((5*A + 4*C)*Sin[c + d*x]^3)/(15*d)}
{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2), x, 2, (1/8)*(4*A + 3*C)*x + ((4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2), x, 2, ((3*A + 2*C)*Sin[c + d*x])/(3*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2), x, 2, A*x + (C*x)/2 + (C*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2), x, 2, (A*ArcTanh[Sin[c + d*x]])/d + (C*Sin[c + d*x])/d}
{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2), x, 2, C*x + (A*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2), x, 2, ((A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2), x, 2, ((2*A + 3*C)*Tan[c + d*x])/(3*d) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + C*Cos[c + d*x]^2), x, 3, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + C*Cos[c + d*x]^2), x, 3, ((4*A + 5*C)*Tan[c + d*x])/(5*d) + (A*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((4*A + 5*C)*Tan[c + d*x]^3)/(15*d)}
{Sec[c + d*x]^7*(A + C*Cos[c + d*x]^2), x, 4, ((5*A + 6*C)*ArcTanh[Sin[c + d*x]])/(16*d) + ((5*A + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((5*A + 6*C)*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (A*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)}
{Sec[c + d*x]^8*(A + C*Cos[c + d*x]^2), x, 3, ((6*A + 7*C)*Tan[c + d*x])/(7*d) + (A*Sec[c + d*x]^6*Tan[c + d*x])/(7*d) + (2*(6*A + 7*C)*Tan[c + d*x]^3)/(21*d) + ((6*A + 7*C)*Tan[c + d*x]^5)/(35*d)}

{Cos[c + d*x]^m*((-C)*((m + 1)/(m + 2)) + C*Cos[c + d*x]^2), x, 1, (C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m))}
{Cos[c + d*x]^m*(A + (-A)*((m + 2)/(m + 1))*Cos[c + d*x]^2), x, 1, -((A*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m)))}


{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 5, (1/8)*a*(4*A + 3*C)*x + (2*a*(5*A + 4*C)*Sin[c + d*x])/(15*d) + (a*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(5*A + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, (1/8)*a*(4*A + 3*C)*x + (a*(3*A + 2*C)*Sin[c + d*x])/(3*d) + (a*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*Cos[c + d*x]^2*Sin[c + d*x])/(12*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 3, (1/2)*a*(2*A + C)*x + (a*(3*A + C)*Sin[c + d*x])/(3*d) - (a*C*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d)}
{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, (1/2)*a*(2*A + C)*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/(2*d) + (C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d)}
{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, a*C*x + (a*A*ArcTanh[Sin[c + d*x]])/d - (a*(A - C)*Sin[c + d*x])/d + (A*(a + a*Cos[c + d*x])*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, a*C*x + (a*(A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*A*Tan[c + d*x])/(2*d) + (A*(a + a*Sec[c + d*x])*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 5, (a*(A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 6, (a*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (a*(3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (A*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, (1/16)*a^2*(14*A + 11*C)*x + (4*a^2*(5*A + 4*C)*Sin[c + d*x])/(15*d) + (a^2*(14*A + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (2*a^2*(5*A + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a^2*(10*A + 9*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, (1/4)*a^2*(4*A + 3*C)*x + (a^2*(4*A + 3*C)*Sin[c + d*x])/(3*d) + (a^2*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(12*d) + ((10*A + 3*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(30*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(10*a*d)}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 4, (1/8)*a^2*(12*A + 7*C)*x + (a^2*(12*A + 7*C)*Sin[c + d*x])/(6*d) + (a^2*(12*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) - (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)}
{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, (2*a^2*A + a^2*C)*x + (a^2*A*ArcTanh[Sin[c + d*x]])/d + (a^2*(A + C)*Sin[c + d*x])/d + (a*C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(3*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, (1/2)*a^2*(2*A + 3*C)*x + (2*a^2*A*ArcTanh[Sin[c + d*x]])/d - (a^2*(2*A - 3*C)*Sin[c + d*x])/(2*d) - (a*(2*A - C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (A*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, 2*a^2*C*x + (a^2*(3*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (a^2*(3*A - 2*C)*Sin[c + d*x])/(2*d) + (a*A*(a + a*Cos[c + d*x])*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, a^2*C*x + (a^2*(A + 2*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(A + C)*Tan[c + d*x])/d + (a*A*(a + a*Sec[c + d*x])*Tan[c + d*x])/(3*d) + (A*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, (a^2*(7*A + 12*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(7*A + 12*C)*Tan[c + d*x])/(6*d) + (a^2*(7*A + 12*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (A*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(6*d)}
{Sec[c + d*x]^6*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 7, (a^2*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(4*d) + (a^2*(18*A + 25*C)*Tan[c + d*x])/(15*d) + (a^2*(3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^2*(9*A + 10*C)*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + (a*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(10*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}


{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, (1/16)*a^3*(26*A + 21*C)*x + (2*a^3*(133*A + 108*C)*Sin[c + d*x])/(105*d) + (a^3*(26*A + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(133*A + 108*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (a^3*(154*A + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (a^2*(A + C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(14*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (1/16)*a^3*(30*A + 23*C)*x + (a^3*(30*A + 23*C)*Sin[c + d*x])/(12*d) + (a^3*(30*A + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(48*d) + (a*(30*A + 23*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(120*d) + ((30*A + 7*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(120*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(10*a*d)}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 5, (1/8)*a^3*(20*A + 13*C)*x + (a^3*(20*A + 13*C)*Sin[c + d*x])/(6*d) + (a^3*(20*A + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*(20*A + 13*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) - (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d)}
{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (1/8)*a^3*(28*A + 15*C)*x + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(4*A + 3*C)*Sin[c + d*x])/(8*d) + (a^2*(4*A + 5*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(8*d) + (a*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(4*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (1/2)*a^3*(6*A + 5*C)*x + (3*a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*C*Sin[c + d*x])/(2*d) - (a^2*(6*A - 5*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d) - (a*(3*A - C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (1/2)*a^3*(2*A + 7*C)*x + (a^3*(7*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*(A - C)*Sin[c + d*x])/(2*d) - (a^2*(4*A - C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (3*a*A*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, 3*a^3*C*x + (a^3*(5*A + 6*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*A*Sin[c + d*x])/(2*d) + (a^2*(5*A + 3*C)*(a + a*Cos[c + d*x])*Tan[c + d*x])/(3*d) + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, a^3*C*x + (a^3*(15*A + 28*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(3*A + 4*C)*Tan[c + d*x])/(8*d) + (a^2*(5*A + 4*C)*(a + a*Sec[c + d*x])*Tan[c + d*x])/(8*d) + (a*A*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(4*d) + (A*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, (a^3*(13*A + 20*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(13*A + 20*C)*Tan[c + d*x])/(6*d) + (a^3*(13*A + 20*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(13*A + 20*C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + (3*A*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d)}
{Sec[c + d*x]^7*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 8, (a^3*(23*A + 30*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(34*A + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(23*A + 30*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(73*A + 90*C)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + (a^2*(31*A + 30*C)*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)}


{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 8, (1/128)*a^4*(392*A + 323*C)*x + (8*a^4*(63*A + 52*C)*Sin[c + d*x])/(105*d) + (a^4*(392*A + 323*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (4*a^4*(63*A + 52*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (a^4*(2408*A + 2007*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2240*d) + (7*a^3*(8*A + 7*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(120*d) + (a^2*(56*A + 61*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(336*d) + (a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(14*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(8*d)}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, (1/4)*a^4*(14*A + 11*C)*x + (a^4*(14*A + 11*C)*Sin[c + d*x])/(3*d) + (a^4*(14*A + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(12*d) + (a^2*(14*A + 11*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(30*d) + (a*(14*A + 11*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(70*d) + ((21*A + 4*C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(105*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(21*a*d)}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 6, (7/16)*a^4*(10*A + 7*C)*x + (7*a^4*(10*A + 7*C)*Sin[c + d*x])/(12*d) + (7*a^4*(10*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(48*d) + (7*a^2*(10*A + 7*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(120*d) + (a*(10*A + 7*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(40*d) - (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d)}
{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, (1/2)*a^4*(12*A + 7*C)*x + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (a^4*(10*A + 7*C)*Sin[c + d*x])/(2*d) + (a^3*(8*A + 7*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (a^2*(5*A + 7*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(15*d) + (a*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)}
{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, (1/8)*a^4*(52*A + 35*C)*x + (4*a^4*A*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(4*A + 7*C)*Sin[c + d*x])/(8*d) - (a^3*(12*A - 35*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(24*d) - (a^2*(12*A - 7*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - (a*(4*A - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^4*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, 2*a^4*(2*A + 3*C)*x + (a^4*(13*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(A - 2*C)*Sin[c + d*x])/(2*d) - (a^3*(9*A - 4*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(3*d) - (a^2*(15*A - 2*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + (2*a*A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, (1/2)*a^4*(2*A + 13*C)*x + (2*a^4*(3*A + 2*C)*ArcTanh[Sin[c + d*x]])/d - (5*a^4*(2*A - C)*Sin[c + d*x])/(2*d) - (a^3*(22*A + 3*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (a^2*(8*A + 3*C)*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/(3*d) + (2*a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, 4*a^4*C*x + (a^4*(35*A + 52*C)*ArcTanh[Sin[c + d*x]])/(8*d) - (5*a^4*(7*A + 4*C)*Sin[c + d*x])/(8*d) + (a^3*(35*A + 36*C)*(a + a*Cos[c + d*x])*Tan[c + d*x])/(12*d) + (a^2*(7*A + 4*C)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, a^4*C*x + (a^4*(7*A + 12*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^4*(7*A + 10*C)*Tan[c + d*x])/(2*d) + (a^3*(7*A + 8*C)*(a + a*Sec[c + d*x])*Tan[c + d*x])/(6*d) + (a^2*(7*A + 5*C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(15*d) + (a*A*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(5*d) + (A*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)}
{Sec[c + d*x]^7*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 8, (7*a^4*(7*A + 10*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (7*a^4*(7*A + 10*C)*Tan[c + d*x])/(12*d) + (7*a^4*(7*A + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(48*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (7*a^2*(7*A + 10*C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(120*d) + (a*(7*A + 10*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(40*d) + (2*A*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(15*d)}
{Sec[c + d*x]^8*(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 9, (a^4*(11*A + 14*C)*ArcTanh[Sin[c + d*x]])/(4*d) + (a^4*(454*A + 581*C)*Tan[c + d*x])/(105*d) + (a^4*(11*A + 14*C)*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^4*(247*A + 308*C)*Sec[c + d*x]^2*Tan[c + d*x])/(210*d) + (a^3*(109*A + 126*C)*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(210*d) + (a^2*(8*A + 7*C)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(35*d) + (2*a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(21*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 5, (3*(4*A + 5*C)*x)/(8*a) - (2*(3*A + 4*C)*Sin[c + d*x])/(3*a*d) + (3*(4*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) - ((3*A + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d) + ((4*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 4, -(((2*A + 3*C)*x)/(2*a)) + (2*(3*A + 4*C)*Sin[c + d*x])/(3*a*d) - ((2*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((3*A + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, ((2*A + 3*C)*x)/(2*a) - ((A + 2*C)*Sin[c + d*x])/(a*d) + ((2*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, -((C*x)/a) + (C*Sin[c + d*x])/(a*d) + ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, (C*x)/a + (A*ArcTanh[Sin[c + d*x]])/(a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 4, -((A*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*A + C)*Tan[c + d*x])/(a*d) - ((A + C)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 5, ((3*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((2*A + C)*Tan[c + d*x])/(a*d) + ((3*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 6, -(((3*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + (2*(4*A + 3*C)*Tan[c + d*x])/(3*a*d) - ((3*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((4*A + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 6, ((28*A + 55*C)*x)/(8*a^2) - (16*(A + 2*C)*Sin[c + d*x])/(3*a^2*d) + ((28*A + 55*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) - (8*(A + 2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + ((28*A + 55*C)*Cos[c + d*x]^3*Sin[c + d*x])/(12*a^2*d) - ((A + C)*Cos[c + d*x]^5*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (2*(A + 2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 5, -(((2*A + 5*C)*x)/a^2) + (2*(5*A + 12*C)*Sin[c + d*x])/(3*a^2*d) - ((2*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(a^2*d) + ((5*A + 12*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (2*(2*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 4, ((2*A + 7*C)*x)/(2*a^2) - (4*(A + 4*C)*Sin[c + d*x])/(3*a^2*d) + ((2*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (2*(A + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 4, -((2*C*x)/a^2) + ((A + 4*C)*Sin[c + d*x])/(3*a^2*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (2*C*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 3, (C*x)/a^2 + ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((A - 5*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 3, (A*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (2*(2*A - C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 5, -((2*A*ArcTanh[Sin[c + d*x]])/(a^2*d)) + ((10*A + C)*Tan[c + d*x])/(3*a^2*d) - ((A + C)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (2*A*Tan[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 6, ((7*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (4*(4*A + C)*Tan[c + d*x])/(3*a^2*d) + ((7*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (2*(4*A + C)*Sec[c + d*x]*Tan[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 7, -(((5*A + 2*C)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + (2*(12*A + 5*C)*Tan[c + d*x])/(3*a^2*d) - ((5*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(a^2*d) + ((12*A + 5*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (2*(5*A + 2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 6, -(((6*A + 23*C)*x)/(2*a^3)) + (8*(9*A + 34*C)*Sin[c + d*x])/(15*a^3*d) - ((6*A + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + (4*(9*A + 34*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^3*d) - ((A + C)*Cos[c + d*x]^5*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((3*A + 13*C)*Cos[c + d*x]^4*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((6*A + 23*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 5, ((2*A + 13*C)*x)/(2*a^3) - (2*(11*A + 76*C)*Sin[c + d*x])/(15*a^3*d) + ((2*A + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((11*A + 76*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 5, -((3*C*x)/a^3) + ((2*A + 27*C)*Sin[c + d*x])/(15*a^3*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A - 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (3*C*Sin[c + d*x])/(a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 4, (C*x)/a^3 - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((3*A - 7*C)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((3*A - 22*C)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 3, ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (2*(A - 4*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*A + 7*C)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 4, (A*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*A - 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((22*A - 3*C)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 6, -((3*A*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (2*(36*A + C)*Tan[c + d*x])/(15*a^3*d) - ((A + C)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*A - C)*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (3*A*Tan[c + d*x])/(a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 7, ((13*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(76*A + 11*C)*Tan[c + d*x])/(15*a^3*d) + ((13*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*A + C)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((76*A + 11*C)*Sec[c + d*x]*Tan[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 8, -(((23*A + 6*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d)) + (8*(34*A + 9*C)*Tan[c + d*x])/(15*a^3*d) - ((23*A + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) + (4*(34*A + 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a^3*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((13*A + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((23*A + 6*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 6, ((2*A + 21*C)*x)/(2*a^4) - (32*(5*A + 54*C)*Sin[c + d*x])/(105*a^4*d) + ((2*A + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((A + C)*Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3) - ((10*A + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - (16*(5*A + 54*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 6, -((4*C*x)/a^4) + (2*(3*A + 122*C)*Sin[c + d*x])/(105*a^4*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + (2*(A - 6*C)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((3*A - 88*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + (4*C*Sin[c + d*x])/(a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 5, (C*x)/a^4 - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + (2*(2*A - 5*C)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((8*A - 55*C)*Cos[c + d*x]*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + (8*(A - 20*C)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 4, -(((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4)) + (2*(3*A - 4*C)*Cos[c + d*x]*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((A - 6*C)*Sin[c + d*x])/(21*a^2*d*(a + a*Cos[c + d*x])^2) + (4*(2*A + 9*C)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 4, ((A + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A - 11*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((6*A + 13*C)*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + ((6*A + 13*C)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 5, (A*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((A + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*(5*A - 2*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - ((55*A - 8*C)*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - (8*(20*A - C)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 7, -((4*A*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (2*(332*A + 3*C)*Tan[c + d*x])/(105*a^4*d) - ((A + C)*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*(6*A - C)*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - ((88*A - 3*C)*Tan[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - (4*A*Tan[c + d*x])/(a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 8, ((21*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (32*(54*A + 5*C)*Tan[c + d*x])/(105*a^4*d) + ((21*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*A*Sec[c + d*x]*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3) - ((129*A + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - (16*(54*A + 5*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 9, -((2*(11*A + 2*C)*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (8*(454*A + 83*C)*Tan[c + d*x])/(105*a^4*d) - (2*(11*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(a^4*d) + (4*(454*A + 83*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*(8*A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - ((178*A + 31*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - (4*(11*A + 2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^3*d*(a + a*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+C Cos[c+d x]^2) (a+a Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 6, (4*a*(99*A + 80*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(99*A + 80*C)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*C*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (8*(99*A + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*C*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (4*(99*A + 80*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*a*d)}
{Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 5, (2*a*(21*A + 16*C)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*C*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) - (4*(21*A + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d) + (2*(21*A + 16*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)}
{Cos[c + d*x]^1*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 4, (2*a*(35*A + 27*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A + 18*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)}
{Cos[c + d*x]^0*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 3, (2*a*(15*A + 7*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (4*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 3, (2*Sqrt[2]*a*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 3, (Sqrt[2]*a*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) - (a*(A - 2*C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d}
{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 3, (a*(3*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 4, (a*(5*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(5*A + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 5, (a*(35*A + 48*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(32*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(35*A + 48*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(35*A + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 6, (2*a^2*(143*A + 112*C)*Sin[c + d*x])/(165*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(33*A + 28*C)*Cos[c + d*x]^3*Sin[c + d*x])/(231*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(143*A + 112*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1155*d) + (2*a*C*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(33*d) + (2*(143*A + 112*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(385*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 5, (8*a^2*(63*A + 47*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(63*A + 47*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(63*A + 22*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(21*a*d)}
{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 4, (8*a^2*(35*A + 19*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(35*A + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (4*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 4, (2*Sqrt[2]*a^2*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(5*A + 4*C)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 4, (3*Sqrt[2]*a^2*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(3*A - 8*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(3*A - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 4, (a^2*(7*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(5*A - 8*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (3*a*A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 4, (a^2*(11*A + 24*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(19*A + 24*C)*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 5, (a^2*(75*A + 112*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(32*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(75*A + 112*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(13*A + 16*C)*Sec[c + d*x]*Tan[c + d*x])/(32*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(8*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 6, (a^2*(133*A + 176*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(64*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(133*A + 176*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(133*A + 176*C)*Sec[c + d*x]*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(67*A + 80*C)*Sec[c + d*x]^2*Tan[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (3*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}


{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 7, (2*a^3*(10439*A + 8368*C)*Sin[c + d*x])/(6435*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2717*A + 2224*C)*Cos[c + d*x]^3*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a^2*(10439*A + 8368*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(45045*d) + (2*a^2*(143*A + 136*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d) + (2*a*(10439*A + 8368*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15015*d) + (10*a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)}
{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 6, (64*a^3*(33*A + 25*C)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(33*A + 25*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a*(33*A + 25*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(231*d) + (2*(99*A + 26*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d) + (10*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*a*d)}
{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 5, (64*a^3*(21*A + 13*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(21*A + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(21*A + 13*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) - (4*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 5, (2*Sqrt[2]*a^3*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(49*A + 32*C)*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(7*A + 8*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 5, (5*Sqrt[2]*a^3*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(15*A + 64*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(15*A - 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (a*(5*A - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Tan[c + d*x])/d}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 5, (a^3*(19*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - (a^3*(27*A - 56*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(21*A - 8*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (5*a*A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 5, (5*a^3*(5*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - (a^3*(49*A - 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(31*A + 24*C)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (5*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 5, (a^3*(163*A + 304*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(32*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(299*A + 432*C)*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(17*A + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d) + (5*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 6, (a^3*(283*A + 400*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(64*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(283*A + 400*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(787*A + 1040*C)*Sec[c + d*x]*Tan[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(79*A + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(240*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7, x, 7, (a^3*(1015*A + 1304*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(256*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1015*A + 1304*C)*Tan[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1015*A + 1304*C)*Sec[c + d*x]*Tan[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(109*A + 136*C)*Sec[c + d*x]^2*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(23*A + 24*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(96*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(12*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]], x, 6, -((2*(A + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + (4*(147*A + 143*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(21*A + 19*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*C*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(21*A + 29*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d)}
{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]], x, 5, (2*(A + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) - (4*(35*A + 37*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*C*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A + 31*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]], x, 4, -((2*(A + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + (2*(15*A + 14*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) - (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]], x, 3, (2*(A + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) - (4*C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + a*Cos[c + d*x]], x, 4, (-2*(A + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]], x, 4, (2*(A + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) - (Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (A*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + a*Cos[c + d*x]], x, 5, (-2*(A + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + ((7*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - (A*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + a*Cos[c + d*x]], x, 6, (2*(A + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) - ((9*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + ((7*A + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - (A*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/Sqrt[a + a*Cos[c + d*x]], x, 7, (-2*(A + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + ((107*A + 112*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(32*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - ((21*A + 16*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + ((43*A + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) - (A*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 6, ((11*A + 19*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((455*A + 799*C)*Sin[c + d*x])/(105*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((35*A + 67*C)*Cos[c + d*x]^2*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((7*A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((245*A + 397*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(210*a^2*d)}
{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 5, -(((7*A + 15*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((15*A + 31*C)*Sin[c + d*x])/(5*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((5*A + 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((5*A + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d)}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 4, ((3*A + 11*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((3*A + 13*C)*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((3*A + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 3, ((A - 7*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*C*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 4, -((5*A - 3*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 5, ((9*A + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - (3*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A + C)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(3/2), x, 6, -((13*A + 5*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((19*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((7*A + 2*C)*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((2*A + C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^(3/2), x, 7, ((17*A + 9*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((47*A + 24*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*a*d*Sqrt[a + a*Cos[c + d*x]]) + (3*(7*A + 4*C)*Tan[c + d*x])/(8*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((13*A + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(12*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 6, -(((75*A + 283*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((5*A + 21*C)*Cos[c + d*x]^3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((465*A + 1729*C)*Sin[c + d*x])/(120*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((45*A + 157*C)*Cos[c + d*x]^2*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((195*A + 787*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*a^3*d)}
{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 5, ((19*A + 163*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((A + 17*C)*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((21*A + 197*C)*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + (5*(3*A + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)}
{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 4, (5*(A - 15*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A - 13*C)*Cos[c + d*x]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((A + 49*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 3, ((3*A + 19*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 5, -((43*A - 5*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((11*A - 5*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 6, ((115*A + 3*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - (5*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((15*A - C)*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((35*A + 3*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(5/2), x, 7, -((219*A + 43*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((39*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((63*A + 11*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((19*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((31*A + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{(A + C*Cos[c + d*x]^2)*Cos[c + d*x]^(5/2), x, 3, (2*(9*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)}
{(A + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 3, (2*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{(A + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 2, (2*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{(A + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 2, (2*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(A + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 2, -((2*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 2, (2*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{(A + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2), x, 3, -((2*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/Cos[c + d*x]^(9/2), x, 3, (2*(5*A + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(5*A + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2))}

{(-1 + 3*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 1, (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d}


{Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 8, (2*a*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*a*(11*A + 9*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (10*a*(11*A + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(11*A + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (4*a*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 7, (2*a*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*a*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (4*a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(9*d)}
{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 6, (2*a*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d)}
{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 5, (2*a*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 5, (-2*a*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a*(3*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 5, (-2*a*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a*A*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 6, (-2*a*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a*A*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, (-2*a*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a*A*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*a*(5*A + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}


{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 8, (4*a^2*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^2*(33*A + 25*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a^2*(33*A + 25*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(99*A + 89*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (8*a*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)}
{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 7, (16*a^2*(3*A + 2*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(21*A + 19*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (8*a*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(63*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)}
{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 6, (4*a^2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*(7*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(35*A + 33*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (8*a*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)}
{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 6, (16*a^2*C*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(15*A - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*a*(5*A - C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 6, (-4*a^2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (8*a^2*(A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(5*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (8*a*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 6, (-16*a^2*A*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(17*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (8*a*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, (-4*a^2*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*(3*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(33*A + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^2*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (8*a*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 8, (-16*a^2*(2*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(19*A + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^2*(5*A + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (16*a^2*(2*A + 3*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (8*a*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}


{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2), x, 9, (4*a^3*(221*A + 175*C)*EllipticE[(c + d*x)/2, 2])/(195*d) + (4*a^3*(121*A + 95*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(121*A + 95*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(221*A + 175*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(585*d) + (40*a^3*(143*A + 118*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*a^2*(143*A + 145*C)*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(1287*d) + (12*a*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d)}
{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2), x, 8, (4*a^3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(143*A + 105*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(143*A + 105*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (8*a^3*(44*A + 35*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(385*d) + (2*a^2*(33*A + 35*C)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(231*d) + (4*a*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(33*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d)}
{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, (4*a^3*(27*A + 17*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(21*A + 11*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(21*A + 16*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a^2*(63*A + 73*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(315*d) + (4*a*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)}
{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 7, (4*a^3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(35*A + 13*C)*EllipticF[(c + d*x)/2, 2])/(21*d) - (4*a^3*(35*A - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (2*a^2*(35*A - 11*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d) - (2*a*(7*A - C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 7, (-4*a^3*(5*A - 9*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (8*a^3*(10*A - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*a^2*(35*A - 3*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (4*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 7, (-4*a^3*(9*A - 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(21*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a^2*(11*A + 5*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (4*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, (-4*a^3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(13*A + 35*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(53*A + 70*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(7*A + 5*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (12*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 8, (-4*a^3*(17*A + 27*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(11*A + 21*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(16*A + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(17*A + 27*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(73*A + 63*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (4*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}
{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 9, (-4*a^3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(105*A + 143*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a^3*(35*A + 44*C)*Sin[c + d*x])/(385*d*Cos[c + d*x]^(5/2)) + (4*a^3*(105*A + 143*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(3/2)) + (4*a^3*(5*A + 7*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(35*A + 33*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (4*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(33*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 7, (-3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(7*A + 9*C)*EllipticF[(c + d*x)/2, 2])/(21*a*d) + (5*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((5*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((7*A + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 6, (3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((5*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 5, -(((A + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])), x, 4, ((A + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((A - C)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])), x, 5, -(((3*A + C)*EllipticE[(c + d*x)/2, 2])/(a*d)) - ((A - C)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((3*A + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])), x, 6, ((3*A + C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - ((3*A + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])), x, 7, (-3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((7*A + 5*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(7*A + 5*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]))}


{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 7, (4*(5*A + 14*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (4*(5*A + 14*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((A + 3*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 6, -(((A + 7*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + (2*(A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (2*(A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 5, (4*C*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((A - 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((A - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2), x, 5, ((A - C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(A + C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2), x, 6, (-4*A*EllipticE[(c + d*x)/2, 2])/(a^2*d) - ((5*A - C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (4*A*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((5*A - C)*Sin[c + d*x])/(3*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2), x, 7, ((7*A + C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(5*A + C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (2*(5*A + C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((7*A + C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((7*A + C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}


{(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 8, (7*(7*A + 33*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A + 63*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((13*A + 63*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + (7*(7*A + 33*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) - ((A + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(A + 6*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A + 63*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 7, -((9*A + 119*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 11*C)*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + ((A + 11*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((9*A + 119*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x]))}
{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 6, -((A - 49*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A - 13*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (2*(A - 4*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A - 13*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x]))}
{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 6, ((A - 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (2*(2*A - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3), x, 6, ((9*A - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A + C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(3*A - 2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((9*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3), x, 7, -((49*A - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((49*A - C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - (2*(4*A - C)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((13*A - C)*Sin[c + d*x])/(6*a^2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3), x, 8, ((119*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((11*A + C)*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + ((11*A + C)*Sin[c + d*x])/(2*a^3*d*Cos[c + d*x]^(3/2)) - ((119*A + 9*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - (2*A*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((119*A + 9*C)*Sin[c + d*x])/(30*a^2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+C Cos[c+d x]^2) (a+a Cos[c+d x])^(n/2)*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 5, (Sqrt[a]*(48*A + 35*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*d) + (a*(48*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(48*A + 35*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)}
{Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 4, (Sqrt[a]*(8*A + 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) + (a*(8*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 3, (Sqrt[a]*(8*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) + (a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)}
{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 3, (Sqrt[a]*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d - (a*(2*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 3, (2*Sqrt[a]*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 3, (2*a*A*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(8*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 4, (2*a*A*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(24*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 5, (2*a*A*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 21*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(16*A + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}


{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 6, (a^(3/2)*(176*A + 133*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(128*d) + (a^2*(176*A + 133*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(176*A + 133*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 67*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (3*a*C*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 5, (a^(3/2)*(112*A + 75*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*d) + (a^2*(112*A + 75*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(16*A + 13*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(32*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)}
{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 4, (a^(3/2)*(24*A + 11*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) + (a^2*(24*A + 19*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)}
{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 4, (a^(3/2)*(8*A + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) - (a^2*(8*A - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(4*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 4, (3*a^(3/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d - (a^2*(8*A - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 4, (2*a^(3/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a^2*(4*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 4, (8*a^2*(19*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(19*A + 35*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (6*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))}
{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 5, (2*a^2*(52*A + 63*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 189*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(136*A + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}
{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 6, (2*a^2*(28*A + 33*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(112*A + 143*C)*Sin[c + d*x])/(385*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(112*A + 143*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(112*A + 143*C)*Sin[c + d*x])/(1155*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(33*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))}


{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 7, (a^(5/2)*(1304*A + 1015*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(512*d) + (a^3*(1304*A + 1015*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1304*A + 1015*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(136*A + 109*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(24*A + 23*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(96*d) + (a*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)}
{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 6, (a^(5/2)*(400*A + 283*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(128*d) + (a^3*(400*A + 283*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1040*A + 787*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 79*C)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d) + (a*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)}
{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 5, (a^(5/2)*(304*A + 163*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*d) + (a^3*(432*A + 299*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(16*A + 17*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + (5*a*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)}
{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 5, (5*a^(5/2)*(8*A + 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) - (a^3*(24*A - 49*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A - 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (a*(6*A - C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 5, (a^(5/2)*(8*A + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) - (a^3*(56*A - 27*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 5, (5*a^(5/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d - (a^3*(64*A + 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 5, (2*a^(5/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a^3*(32*A + 49*C)*Sin[c + d*x])/(21*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 5, (64*a^3*(13*A + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(13*A + 21*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)) + (2*a*(13*A + 21*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (10*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))}
{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 6, (2*a^3*(232*A + 297*C)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(568*A + 759*C)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(568*A + 759*C)*Sin[c + d*x])/(693*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))}
{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2), x, 7, (2*a^3*(2224*A + 2717*C)*Sin[c + d*x])/(9009*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(15015*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45045*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d*Cos[c + d*x]^(9/2)) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Cos[c + d*x]^(11/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Cos[c + d*x]^(13/2))}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]], x, 6, -((8*A + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*Sqrt[a]*d) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((8*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]], x, 5, ((8*A + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]), x, 4, -((C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]), x, 4, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]), x, 3, (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]), x, 4, -((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(13*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + a*Cos[c + d*x]]), x, 5, (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(43*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2), x, 6, ((8*A + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*a^(3/2)*d) - ((5*A + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((2*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((A + 2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2), x, 5, (-3*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(3/2)*d) + ((A + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)), x, 4, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(3/2)*d) + ((3*A - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)), x, 3, -((7*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((5*A + C)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)), x, 4, ((11*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((7*A + 3*C)*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((19*A + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])^(3/2)), x, 5, -((15*A + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)) + ((9*A + 5*C)*Sin[c + d*x])/(10*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - ((13*A + 5*C)*Sin[c + d*x])/(10*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + ((49*A + 25*C)*Sin[c + d*x])/(10*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2), x, 6, (-5*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(5/2)*d) + ((3*A + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((A - 15*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2), x, 5, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(5/2)*d) + ((5*A - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A - 11*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)), x, 3, ((19*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((9*A - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)), x, 4, (-5*(15*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 3*C)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((49*A + C)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)), x, 5, ((163*A + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((17*A + C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (5*(19*A + 3*C)*Sin[c + d*x])/(48*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((299*A + 27*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+C Cos[c+d x]^2) (a+a Cos[c+d x])^(n/3)*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(2/3), x, 3, (1/80)*a*(40*A + 19*C)*Int[1/(a + a*Cos[c + d*x])^(1/3), x] + (3*a*(40*A + 19*C)*Sin[c + d*x])/(80*d*(a + a*Cos[c + d*x])^(1/3)) - (9*C*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(40*d) + (3*C*(a + a*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*a*d)}
{(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/3), x, 3, (-(1/28))*a*(28*A + 13*C)*Int[1/(a + a*sin[c + Pi/2 + d*x])^(2/3), x] + (3*a*(28*A + 13*C)*Sin[c + d*x])/(28*d*(a + a*Cos[c + d*x])^(2/3)) - (9*C*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(28*d) + (3*C*(a + a*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*a*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/3), x, 2, (1/10)*(10*A + 7*C)*Int[1/(a + a*Cos[c + d*x])^(1/3), x] - (9*C*Sin[c + d*x])/(10*d*(a + a*Cos[c + d*x])^(1/3)) + (3*C*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*a*d)}
{(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(2/3), x, 2, (1/4)*(4*A + 7*C)*Int[1/(a + a*sin[c + Pi/2 + d*x])^(2/3), x] - (9*C*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(2/3)) + (3*C*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*a*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Sec[c+d x]^(m/2) (A+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 4, (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(5*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 4, -((2*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 3, (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 3, -((2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2), x, 3, (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/Sec[c + d*x]^(1/2), x, 3, (2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))}
{(A + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 4, (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/Sec[c + d*x]^(5/2), x, 4, (2*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*C*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2))}


{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, (-2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (4*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, (-2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, (-2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 6, (-2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a*(3*A - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 6, (2*a*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a*C*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 7, (2*a*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a*C*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 8, (2*a*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a*C*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^(5/2)) + (2*a*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 9, (-16*a^2*(2*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (16*a^2*(2*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^2*(5*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(19*A + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (8*a*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, (-4*a^2*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*(3*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(33*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (8*a*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, (-16*a^2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(17*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (8*a*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 7, (-4*a^2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*(A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(5*A - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (8*a*A*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 7, (16*a^2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(15*A - 7*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*a*(5*A - C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 7, (4*a^2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*(7*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(35*A + 33*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (8*a*C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 8, (16*a^2*(3*A + 2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(21*A + 19*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (8*a*C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (4*a^2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 9, (4*a^2*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a^2*(33*A + 25*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(99*A + 89*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (8*a*C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (4*a^2*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (8*a^2*(33*A + 25*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 10, (-4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(105*A + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(5*A + 7*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(105*A + 143*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (8*a^3*(35*A + 44*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(385*d) + (2*a^2*(35*A + 33*C)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (4*a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(33*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)}
{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 9, (-4*a^3*(17*A + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(17*A + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (8*a^3*(16*A + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a^2*(73*A + 63*C)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (4*a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, (-4*a^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(53*A + 70*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a^2*(7*A + 5*C)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (12*a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 8, (-4*a^3*(9*A - 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(21*A + 5*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(11*A + 5*C)*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 8, (-4*a^3*(5*A - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (8*a^3*(10*A - 3*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*a^2*(35*A - 3*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (4*a*A*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 8, (4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(35*A + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a^3*(35*A - 41*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) - (2*a^2*(35*A - 11*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]) - (2*a*(7*A - C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 8, (4*a^3*(27*A + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(21*A + 16*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(63*A + 73*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]) + (4*a*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 9, (4*a^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(143*A + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (8*a^3*(44*A + 35*C)*Sin[c + d*x])/(385*d*Sec[c + d*x]^(3/2)) + (2*a^2*(33*A + 35*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(231*d*Sec[c + d*x]^(3/2)) + (4*a*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(33*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(3/2)) + (4*a^3*(143*A + 105*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 10, (4*a^3*(221*A + 175*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (4*a^3*(121*A + 95*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (40*a^3*(143*A + 118*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*a^2*(143*A + 145*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(5/2)) + (12*a*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(143*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(5/2)) + (4*a^3*(221*A + 175*C)*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (4*a^3*(121*A + 95*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x]), x, 8, (-3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(7*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) - ((5*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((7*A + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x]), x, 7, ((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((3*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((5*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x]), x, 6, -(((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) - ((A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x]), x, 5, ((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]), x, 6, -(((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 7, (3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((5*A + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - ((3*A + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)), x, 8, (-3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)) + ((7*A + 9*C)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - ((5*A + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(7*A + 9*C)*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]])}


{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^2, x, 8, ((7*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(5*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((7*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*(5*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((7*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^2, x, 7, (-4*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - ((5*A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (4*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((5*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^2, x, 6, ((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A - C)*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]), x, 6, (4*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((A - 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A - 5*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)), x, 7, -(((A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*(A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((A + 7*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + (2*(A + 5*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)), x, 8, (4*(5*A + 14*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((A + 3*C)*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (4*(5*A + 14*C)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(A + 3*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])}


{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^3, x, 9, ((119*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((11*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - ((119*A + 9*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((11*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((119*A + 9*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x]))}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^3, x, 8, -((49*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((49*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(4*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x]))}
{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^3, x, 7, ((9*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]) - (2*(3*A - 2*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((9*A - C)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]), x, 7, ((A - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)) + (2*(2*A - 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A - 9*C)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)), x, 7, -((A - 49*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A - 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)) + (2*(A - 4*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A - 13*C)*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)), x, 8, -((9*A + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)) - (2*C*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((9*A + 119*C)*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((A + 11*C)*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)), x, 9, (7*(7*A + 33*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A + 63*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)) - (2*(A + 6*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((13*A + 63*C)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (7*(7*A + 33*C)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((13*A + 63*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Sec[c+d x]^(m/2) (A+C Cos[c+d x]^2) (a+a Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 5, (16*a*(16*A + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 4, (4*a*(24*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 3, (2*a*(8*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 4, (2*Sqrt[a]*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 4, (Sqrt[a]*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a*(2*A - C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 4, (Sqrt[a]*(8*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*C*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])}
{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 5, (Sqrt[a]*(8*A + 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*C*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (a*(8*A + 5*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 6, (Sqrt[a]*(48*A + 35*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a*C*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (a*(48*A + 35*C)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(48*A + 35*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 6, (16*a^2*(112*A + 143*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(112*A + 143*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(112*A + 143*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(385*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(28*A + 33*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(33*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 5, (4*a^2*(136*A + 189*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 189*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 63*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 4, (8*a^2*(19*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(19*A + 35*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (6*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 5, (2*a^(3/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(4*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 5, (3*a^(3/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^2*(8*A - 3*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 5, (a^(3/2)*(8*A + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^2*(8*A - 5*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a*(4*A - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 5, (a^(3/2)*(24*A + 11*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(24*A + 19*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 6, (a^(3/2)*(112*A + 75*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(16*A + 13*C)*Sin[c + d*x])/(32*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)) + (a^2*(112*A + 75*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 7, (a^(3/2)*(176*A + 133*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^2*(80*A + 67*C)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (3*a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(5/2)) + (a^2*(176*A + 133*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(176*A + 133*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(15/2), x, 7, (16*a^3*(8368*A + 10439*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 10439*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 10439*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2224*A + 2717*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(1287*d) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(143*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(13/2)*Sin[c + d*x])/(13*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 6, (4*a^3*(568*A + 759*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(568*A + 759*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(232*A + 297*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 5, (64*a^3*(13*A + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(13*A + 21*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*a*(13*A + 21*C)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (10*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 6, (2*a^(5/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(32*A + 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 6, (5*a^(5/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^3*(64*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*(8*A + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, (a^(5/2)*(8*A + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^3*(56*A - 27*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 6, (5*a^(5/2)*(8*A + 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) - (a^3*(24*A - 49*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A - 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) - (a*(6*A - C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 6, (a^(5/2)*(304*A + 163*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(432*A + 299*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*(16*A + 17*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + (5*a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 7, (a^(5/2)*(400*A + 283*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 787*C)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(80*A + 79*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d*Sec[c + d*x]^(3/2)) + (a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (a^3*(400*A + 283*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 8, (a^(5/2)*(1304*A + 1015*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(512*d) + (a^3*(136*A + 109*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*(24*A + 23*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(96*d*Sec[c + d*x]^(5/2)) + (a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Sec[c + d*x]^(5/2)) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2))/Sqrt[a + a*Cos[c + d*x]], x, 7, -((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(257*A + 273*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(29*A + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(19*A + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + a*Cos[c + d*x]], x, 6, (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + a*Cos[c + d*x]], x, 5, -((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + a*Cos[c + d*x]], x, 4, (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + a*Cos[c + d*x]], x, 5, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]], x, 5, -((C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]), x, 6, ((8*A + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - (C*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 7, -((8*A + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) - (C*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((8*A + 7*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2), x, 7, ((19*A + 11*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((1201*A + 665*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((397*A + 245*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((67*A + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((11*A + 7*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(3/2), x, 6, -((15*A + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((49*A + 25*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((13*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(3/2), x, 5, ((11*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((19*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((7*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(3/2), x, 4, -((7*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(3/2), x, 5, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((3*A - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]), x, 6, (-3*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((A + 3*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)), x, 7, ((8*A + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d) - ((5*A + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((A + 2*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((2*A + 7*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(5/2), x, 7, -((283*A + 75*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((2671*A + 735*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((787*A + 195*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((21*A + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((157*A + 45*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(5/2), x, 6, ((163*A + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((299*A + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((17*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (5*(19*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(5/2), x, 5, (-5*(15*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((49*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(5/2), x, 4, ((19*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((9*A - 7*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]), x, 6, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((5*A - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((5*A - 11*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 7, (-5*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((A - 15*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((3*A + 35*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)), x, 8, ((8*A + 39*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(5/2)*d) - ((43*A + 219*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)) - ((3*A + 19*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((7*A + 31*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((11*A + 63*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


(* ::Section:: *)
(*Cos[c+d x]^m (B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, (3*C*x)/8 + (2*B*Sin[c + d*x])/(3*d) + (3*C*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, (B*x)/2 + (2*C*Sin[c + d*x])/(3*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, (C*x)/2 + (B*Sin[c + d*x])/d + (C*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 2, B*x + (C*Sin[c + d*x])/d}
{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, C*x + (B*ArcTanh[Sin[c + d*x]])/d}
{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, (C*ArcTanh[Sin[c + d*x]])/d + (B*Tan[c + d*x])/d}
{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, (B*ArcTanh[Sin[c + d*x]])/(2*d) + (C*Tan[c + d*x])/d + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 6, (C*ArcTanh[Sin[c + d*x]])/(2*d) + (2*B*Tan[c + d*x])/(3*d) + (C*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^6*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (3*B*ArcTanh[Sin[c + d*x]])/(8*d) + (2*C*Tan[c + d*x])/(3*d) + (3*B*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 6, (3/8)*a*(B + C)*x + (2*a*(5*B + 4*C)*Sin[c + d*x])/(15*d) + (3*a*(B + C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(5*B + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a*(B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 5, (1/8)*a*(4*B + 3*C)*x + (2*a*(B + C)*Sin[c + d*x])/(3*d) + (a*(4*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 3, (1/2)*a*(B + C)*x + (a*(3*B + C)*Sin[c + d*x])/(3*d) + (a*(3*B - C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d)}
{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 3, (1/2)*a*(2*B + C)*x + (a*(B + C)*Sin[c + d*x])/d + (a*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, (a*B + a*C)*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/d}
{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, a*C*x + (a*(B + C)*ArcTanh[Sin[c + d*x]])/d + (a*B*Tan[c + d*x])/d}
{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 5, (a*(B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(B + C)*Tan[c + d*x])/d + (a*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 6, (a*(B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*B + 3*C)*Tan[c + d*x])/(3*d) + (a*(B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^6*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 7, (a*(3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*(B + C)*Tan[c + d*x])/(3*d) + (a*(3*B + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, (1/8)*a^2*(7*B + 6*C)*x + (2*a^2*(10*B + 9*C)*Sin[c + d*x])/(15*d) + (a^2*(7*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(10*B + 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a^2*(5*B + 6*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 4, (1/8)*a^2*(8*B + 7*C)*x + (a^2*(8*B + 7*C)*Sin[c + d*x])/(6*d) + (a^2*(8*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*B - C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)}
{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 4, (1/2)*a^2*(3*B + 2*C)*x + (2*a^2*(3*B + 2*C)*Sin[c + d*x])/(3*d) + (a^2*(3*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, (1/2)*a^2*(4*B + 3*C)*x + (a^2*B*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*B + 3*C)*Sin[c + d*x])/(2*d) + (a*C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d)}
{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, (a^2*B + 2*a^2*C)*x + (a^2*(2*B + C)*ArcTanh[Sin[c + d*x]])/d - (a^2*(B - C)*Sin[c + d*x])/d + (a*B*(a + a*Cos[c + d*x])*Tan[c + d*x])/d}
{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, a^2*C*x + (a^2*(3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(3*B + 2*C)*Tan[c + d*x])/(2*d) + (a*B*(a + a*Sec[c + d*x])*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, (a^2*(2*B + 3*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a^2*(2*B + 3*C)*Tan[c + d*x])/(3*d) + (a^2*(2*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (B*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^6*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 7, (a^2*(7*B + 8*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(4*B + 5*C)*Tan[c + d*x])/(3*d) + (a^2*(7*B + 8*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(5*B + 4*C)*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (a*B*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^7*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 8, (a^2*(6*B + 7*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a^2*(9*B + 10*C)*Tan[c + d*x])/(15*d) + (a^2*(6*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(9*B + 10*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a^2*(6*B + 5*C)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*B*(a + a*Cos[c + d*x])*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}


{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, (1/16)*a^3*(26*B + 23*C)*x + (2*a^3*(19*B + 17*C)*Sin[c + d*x])/(15*d) + (a^3*(26*B + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(19*B + 17*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a^3*(22*B + 21*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a^2*(3*B + 4*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 5, (1/8)*a^3*(15*B + 13*C)*x + (a^3*(15*B + 13*C)*Sin[c + d*x])/(6*d) + (a^3*(15*B + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*(15*B + 13*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5*B - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d)}
{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 5, (5/8)*a^3*(4*B + 3*C)*x + (5*a^3*(4*B + 3*C)*Sin[c + d*x])/(6*d) + (5*a^3*(4*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*(4*B + 3*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (1/2)*a^3*(7*B + 5*C)*x + (a^3*B*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(B + C)*Sin[c + d*x])/(2*d) + (a^2*(3*B + 5*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (a*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (1/2)*a^3*(6*B + 7*C)*x + (a^3*(3*B + C)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*C*Sin[c + d*x])/(2*d) - (a^2*(2*B - C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (a*B*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d}
{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (a^3*B + 3*a^3*C)*x + (a^3*(7*B + 6*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*B*Sin[c + d*x])/(2*d) + (a^2*(2*B + C)*(a + a*Cos[c + d*x])*Tan[c + d*x])/d + (a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, a^3*C*x + (a^3*(5*B + 7*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(B + C)*Tan[c + d*x])/(2*d) + (a^2*(5*B + 3*C)*(a + a*Sec[c + d*x])*Tan[c + d*x])/(6*d) + (a*B*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^6*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, (5*a^3*(3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(3*B + 4*C)*Tan[c + d*x])/(6*d) + (5*a^3*(3*B + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (a*(3*B + 4*C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (B*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^7*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 8, (a^3*(13*B + 15*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(38*B + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(13*B + 15*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^3*(43*B + 45*C)*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + (a^2*(7*B + 5*C)*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 5, (3*(B - C)*x)/(2*a) - (2*(3*B - 4*C)*Sin[c + d*x])/(3*a*d) + (3*(B - C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((3*B - 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d) + ((B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 4, -(((2*B - 3*C)*x)/(2*a)) + (2*(B - C)*Sin[c + d*x])/(a*d) - ((2*B - 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 4, ((B - C)*x)/a + (C*Sin[c + d*x])/(a*d) - ((B - C)*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x]))}
{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, (C*x)/a + ((B - C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, (B*ArcTanh[Sin[c + d*x]])/(a*d) - ((B - C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 5, -(((B - C)*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*B - C)*Tan[c + d*x])/(a*d) - ((B - C)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 6, ((3*B - 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - (2*(B - C)*Tan[c + d*x])/(a*d) + ((3*B - 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((B - C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 7, -((3*(B - C)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + (2*(4*B - 3*C)*Tan[c + d*x])/(3*a*d) - (3*(B - C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((4*B - 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d) - ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 6, ((7*B - 10*C)*x)/(2*a^2) - (8*(2*B - 3*C)*Sin[c + d*x])/(3*a^2*d) + ((7*B - 10*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (4*(2*B - 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + ((B - C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((7*B - 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 5, -(((4*B - 7*C)*x)/(2*a^2)) + (2*(5*B - 8*C)*Sin[c + d*x])/(3*a^2*d) - ((4*B - 7*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + ((B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((5*B - 8*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 5, ((B - 2*C)*x)/a^2 - ((B - 4*C)*Sin[c + d*x])/(3*a^2*d) + ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((B - 2*C)*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 3, (C*x)/a^2 - ((B - C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((2*B - 5*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 3, ((B - C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((B + 2*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 4, (B*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((B - C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((4*B - C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 6, -(((2*B - C)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + (2*(5*B - 2*C)*Tan[c + d*x])/(3*a^2*d) - ((B - C)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((2*B - C)*Tan[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 7, ((7*B - 4*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*(8*B - 5*C)*Tan[c + d*x])/(3*a^2*d) + ((7*B - 4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((B - C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((8*B - 5*C)*Sec[c + d*x]*Tan[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 6, -(((6*B - 13*C)*x)/(2*a^3)) + (8*(9*B - 19*C)*Sin[c + d*x])/(15*a^3*d) - ((6*B - 13*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + ((B - C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((6*B - 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (4*(9*B - 19*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 6, ((B - 3*C)*x)/a^3 - ((7*B - 27*C)*Sin[c + d*x])/(15*a^3*d) + ((B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*B - 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((B - 3*C)*Sin[c + d*x])/(a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 5, (C*x)/a^3 + ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*B - 7*C)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (2*(B - 11*C)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 3, -(((B - C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3)) + ((3*B - 8*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((3*B + 7*C)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 4, ((B - C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*B + 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*B + 3*C)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 5, (B*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((B - C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*B - 2*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (2*(11*B - C)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 7, -(((3*B - C)*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (2*(36*B - 11*C)*Tan[c + d*x])/(15*a^3*d) - ((B - C)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*B - 4*C)*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((3*B - C)*Tan[c + d*x])/(a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 8, ((13*B - 6*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (8*(19*B - 9*C)*Tan[c + d*x])/(15*a^3*d) + ((13*B - 6*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((B - C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*B - 6*C)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (4*(19*B - 9*C)*Sec[c + d*x]*Tan[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*a*(5*B + 7*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*B - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)}


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (8*a^2*(21*B + 19*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(21*B + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*B - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)}


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (64*a^3*(15*B + 13*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(15*B + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(15*B + 13*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (2*(9*B - 2*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 3, -((2*(B - C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + (2*(3*B - 2*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)}


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 3, ((3*B - 7*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((B - C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*C*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])}


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 3, ((5*B + 19*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((B - C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*B - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 6, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (10*C*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 5, (6*C*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 4, (2*B*EllipticE[(1/2)*(c + d*x), 2])/d + (2*C*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 4, (2*C*EllipticE[(1/2)*(c + d*x), 2])/d + (2*B*EllipticF[(1/2)*(c + d*x), 2])/d}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 5, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*C*EllipticF[(1/2)*(c + d*x), 2])/d + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2), x, 6, -((2*C*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(9/2), x, 7, -((6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*C*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*B*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])}


(* ::Subsubsection:: *)
(*n<0*)


(* ::Subsection:: *)
(*Integrands of the form Cos[c+d x]^(m/2 (B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^(n/2)*)
(**)


(* ::Subsection::Closed:: *)
(*Integrands of the form Sec[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, -((6*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, -((2*C*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, -((2*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 5, (2*C*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2), x, 5, (2*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(1/2), x, 6, (6*C*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 7, (6*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (10*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (10*C*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}


(* ::Subsubsection:: *)
(*n<0*)


(* ::Subsection:: *)
(*Integrands of the form Sec[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^(n/2)*)


(* ::Section:: *)
(*Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 6, (1/16)*(6*A + 5*C)*x + (8*B*Sin[c + d*x])/(15*d) + ((6*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (4*B*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + ((6*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) + (C*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, (3*B*x)/8 + (2*(5*A + 4*C)*Sin[c + d*x])/(15*d) + (3*B*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((5*A + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, (1/8)*(4*A + 3*C)*x + (2*B*Sin[c + d*x])/(3*d) + ((4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, (B*x)/2 + ((3*A + 2*C)*Sin[c + d*x])/(3*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, A*x + (C*x)/2 + (B*Sin[c + d*x])/d + (C*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, B*x + (A*ArcTanh[Sin[c + d*x]])/d + (C*Sin[c + d*x])/d}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, C*x + (B*ArcTanh[Sin[c + d*x]])/d + (A*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, ((A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (B*Tan[c + d*x])/d + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, (B*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*A + 3*C)*Tan[c + d*x])/(3*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 6, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (2*B*Tan[c + d*x])/(3*d) + ((3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (3*B*ArcTanh[Sin[c + d*x]])/(8*d) + (2*(4*A + 5*C)*Tan[c + d*x])/(15*d) + (3*B*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((4*A + 5*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (A*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}


{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 5, (1/8)*a*(4*A + 3*B + 3*C)*x + (2*a*(5*A + 5*B + 4*C)*Sin[c + d*x])/(15*d) + (a*(4*A + 3*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(5*A + 5*B + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a*(5*B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, (1/8)*a*(4*A + 4*B + 3*C)*x + (a*(3*A + 2*B + 2*C)*Sin[c + d*x])/(3*d) + (a*(4*A + 4*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(4*B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(12*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 3, (1/2)*a*(2*A + B + C)*x + (a*(3*A + 3*B + C)*Sin[c + d*x])/(3*d) + (a*(3*B - C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, (1/2)*a*(2*A + 2*B + C)*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*(2*B + C)*Sin[c + d*x])/(2*d) + (C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, (a*B + a*C)*x + (a*(A + B)*ArcTanh[Sin[c + d*x]])/d - (a*(A - C)*Sin[c + d*x])/d + (A*(a + a*Cos[c + d*x])*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, a*C*x + (a*(A + 2*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(A + 2*B)*Tan[c + d*x])/(2*d) + (A*(a + a*Sec[c + d*x])*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 5, (a*(A + B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A + 3*B + 3*C)*Tan[c + d*x])/(3*d) + (a*(A + 3*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 6, (a*(3*A + 4*B + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(2*A + 2*B + 3*C)*Tan[c + d*x])/(3*d) + (a*(3*A + 4*B + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(A + 4*B)*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (A*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, (1/16)*a^2*(14*A + 12*B + 11*C)*x + (2*a^2*(10*A + 9*B + 8*C)*Sin[c + d*x])/(15*d) + (a^2*(14*A + 12*B + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(10*A + 9*B + 8*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a^2*(10*A + 12*B + 9*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*(3*B + C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, (1/8)*a^2*(8*A + 7*B + 6*C)*x + (a^2*(8*A + 7*B + 6*C)*Sin[c + d*x])/(6*d) + (a^2*(8*A + 7*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((20*A - 5*B + 6*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d) + ((5*B + 2*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 4, (1/8)*a^2*(12*A + 8*B + 7*C)*x + (a^2*(12*A + 8*B + 7*C)*Sin[c + d*x])/(6*d) + (a^2*(12*A + 8*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*B - C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, (1/2)*a^2*(4*A + 3*B + 2*C)*x + (a^2*A*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*A + 3*B + 2*C)*Sin[c + d*x])/(2*d) + (a*(3*B + 2*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, (1/2)*a^2*(2*A + 4*B + 3*C)*x + (a^2*(2*A + B)*ArcTanh[Sin[c + d*x]])/d - (a^2*(2*A - 2*B - 3*C)*Sin[c + d*x])/(2*d) - (a*(2*A - C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (A*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, (a^2*B + 2*a^2*C)*x + (a^2*(3*A + 4*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (a^2*(3*A + 2*B - 2*C)*Sin[c + d*x])/(2*d) + (a*(A + B)*(a + a*Cos[c + d*x])*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, a^2*C*x + (a^2*(2*A + 3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*A + 3*B + 2*C)*Tan[c + d*x])/(2*d) + (a*(2*A + 3*B)*(a + a*Sec[c + d*x])*Tan[c + d*x])/(6*d) + (A*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, (a^2*(7*A + 8*B + 12*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(7*A + 8*B + 12*C)*Tan[c + d*x])/(6*d) + (a^2*(7*A + 8*B + 12*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((A + 2*B)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(6*d)}
{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 7, (a^2*(6*A + 7*B + 8*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(18*A + 20*B + 25*C)*Tan[c + d*x])/(15*d) + (a^2*(6*A + 7*B + 8*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(18*A + 25*B + 20*C)*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + (a*(2*A + 5*B)*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}


{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, (1/16)*a^3*(26*A + 23*B + 21*C)*x + (2*a^3*(133*A + 119*B + 108*C)*Sin[c + d*x])/(105*d) + (a^3*(26*A + 23*B + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(133*A + 119*B + 108*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (a^3*(154*A + 147*B + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (a^2*(3*A + 4*B + 3*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (a*(7*B + 3*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(42*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (1/16)*a^3*(30*A + 26*B + 23*C)*x + (a^3*(30*A + 26*B + 23*C)*Sin[c + d*x])/(12*d) + (a^3*(30*A + 26*B + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(48*d) + (a*(30*A + 26*B + 23*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(120*d) + ((30*A - 6*B + 7*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(120*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(6*d) + ((2*B + C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(10*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 5, (1/8)*a^3*(20*A + 15*B + 13*C)*x + (a^3*(20*A + 15*B + 13*C)*Sin[c + d*x])/(6*d) + (a^3*(20*A + 15*B + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*(20*A + 15*B + 13*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5*B - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (1/8)*a^3*(28*A + 20*B + 15*C)*x + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(4*A + 4*B + 3*C)*Sin[c + d*x])/(8*d) + (a^2*(12*A + 20*B + 15*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(24*d) + (a*(4*B + 3*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (1/2)*a^3*(6*A + 7*B + 5*C)*x + (a^3*(3*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(B + C)*Sin[c + d*x])/(2*d) - (a^2*(6*A - 3*B - 5*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d) - (a*(3*A - C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (1/2)*a^3*(2*A + 6*B + 7*C)*x + (a^3*(7*A + 6*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*(A - C)*Sin[c + d*x])/(2*d) - (a^2*(4*A + 2*B - C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (a*(3*A + 2*B)*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, (a^3*B + 3*a^3*C)*x + (a^3*(5*A + 7*B + 6*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*(A + B)*Sin[c + d*x])/(2*d) + (a^2*(5*A + 6*B + 3*C)*(a + a*Cos[c + d*x])*Tan[c + d*x])/(3*d) + (a*(A + B)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 6, a^3*C*x + (a^3*(15*A + 20*B + 28*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(3*A + 4*B + 4*C)*Tan[c + d*x])/(8*d) + (a^2*(15*A + 20*B + 12*C)*(a + a*Sec[c + d*x])*Tan[c + d*x])/(24*d) + (a*(3*A + 4*B)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (A*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, (a^3*(13*A + 15*B + 20*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(13*A + 15*B + 20*C)*Tan[c + d*x])/(6*d) + (a^3*(13*A + 15*B + 20*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(13*A + 15*B + 20*C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + ((3*A + 5*B)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d)}
{Sec[c + d*x]^7*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 8, (a^3*(23*A + 26*B + 30*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(34*A + 38*B + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(23*A + 26*B + 30*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(73*A + 86*B + 90*C)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + (a^2*(31*A + 42*B + 30*C)*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (a*(A + 2*B)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)}


{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 8, (1/128)*a^4*(392*A + 352*B + 323*C)*x + (2*a^4*(252*A + 227*B + 208*C)*Sin[c + d*x])/(105*d) + (a^4*(392*A + 352*B + 323*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^4*(252*A + 227*B + 208*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (a^4*(2408*A + 2208*B + 2007*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2240*d) + (7*a^3*(8*A + 8*B + 7*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])*Sin[c + d*x])/(120*d) + (a^2*(56*A + 80*B + 61*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(336*d) + (a*(2*B + C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(14*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(8*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, (1/16)*a^4*(56*A + 49*B + 44*C)*x + (a^4*(56*A + 49*B + 44*C)*Sin[c + d*x])/(12*d) + (a^4*(56*A + 49*B + 44*C)*Cos[c + d*x]*Sin[c + d*x])/(48*d) + (a^2*(56*A + 49*B + 44*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(120*d) + (a*(56*A + 49*B + 44*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(280*d) + ((42*A - 7*B + 8*C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(210*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + ((7*B + 4*C)*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(42*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 6, (7/16)*a^4*(10*A + 8*B + 7*C)*x + (7*a^4*(10*A + 8*B + 7*C)*Sin[c + d*x])/(12*d) + (7*a^4*(10*A + 8*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(48*d) + (7*a^2*(10*A + 8*B + 7*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(120*d) + (a*(10*A + 8*B + 7*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(40*d) + ((6*B - C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, (1/8)*a^4*(48*A + 35*B + 28*C)*x + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (a^4*(40*A + 35*B + 28*C)*Sin[c + d*x])/(8*d) + (a^3*(32*A + 35*B + 28*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(24*d) + (a^2*(20*A + 35*B + 28*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (a*(5*B + 4*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, (1/8)*a^4*(52*A + 48*B + 35*C)*x + (a^4*(4*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(4*A + 8*B + 7*C)*Sin[c + d*x])/(8*d) - (a^3*(12*A - 32*B - 35*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(24*d) - (a^2*(12*A - 4*B - 7*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - (a*(4*A - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^4*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, (1/2)*a^4*(8*A + 13*B + 12*C)*x + (a^4*(13*A + 8*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(A - B - 2*C)*Sin[c + d*x])/(2*d) - (a^3*(18*A + 3*B - 8*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d) - (a^2*(15*A + 6*B - 2*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + (a*(2*A + B)*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, (1/2)*a^4*(2*A + 8*B + 13*C)*x + (a^4*(12*A + 13*B + 8*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(2*A + B - C)*Sin[c + d*x])/(2*d) - (a^3*(22*A + 18*B + 3*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (a^2*(16*A + 15*B + 6*C)*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/(6*d) + (a*(4*A + 3*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, (a^4*B + 4*a^4*C)*x + (a^4*(35*A + 48*B + 52*C)*ArcTanh[Sin[c + d*x]])/(8*d) - (5*a^4*(7*A + 8*B + 4*C)*Sin[c + d*x])/(8*d) + (a^3*(35*A + 44*B + 36*C)*(a + a*Cos[c + d*x])*Tan[c + d*x])/(12*d) + (a^2*(7*A + 8*B + 4*C)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(A + B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 7, a^4*C*x + (a^4*(28*A + 35*B + 48*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^4*(28*A + 35*B + 40*C)*Tan[c + d*x])/(8*d) + (a^3*(28*A + 35*B + 32*C)*(a + a*Sec[c + d*x])*Tan[c + d*x])/(24*d) + (a^2*(28*A + 35*B + 20*C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + (a*(4*A + 5*B)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (A*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)}
{Sec[c + d*x]^7*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 8, (7*a^4*(7*A + 8*B + 10*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (7*a^4*(7*A + 8*B + 10*C)*Tan[c + d*x])/(12*d) + (7*a^4*(7*A + 8*B + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(48*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (7*a^2*(7*A + 8*B + 10*C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(120*d) + (a*(7*A + 8*B + 10*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(40*d) + ((2*A + 3*B)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(15*d)}
{Sec[c + d*x]^8*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 9, (a^4*(44*A + 49*B + 56*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^4*(454*A + 504*B + 581*C)*Tan[c + d*x])/(105*d) + (a^4*(44*A + 49*B + 56*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^4*(988*A + 1113*B + 1232*C)*Sec[c + d*x]^2*Tan[c + d*x])/(840*d) + (a^3*(436*A + 511*B + 504*C)*(a + a*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(840*d) + (a^2*(16*A + 21*B + 14*C)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(70*d) + (a*(4*A + 7*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(42*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 5, (3*(4*A - 4*B + 5*C)*x)/(8*a) - (2*(3*A - 4*B + 4*C)*Sin[c + d*x])/(3*a*d) + (3*(4*A - 4*B + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) - ((3*A - 4*B + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d) + ((4*A - 4*B + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 4, -(((2*A - 3*B + 3*C)*x)/(2*a)) + (2*(3*A - 3*B + 4*C)*Sin[c + d*x])/(3*a*d) - ((2*A - 3*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((3*A - 3*B + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, ((2*A - 2*B + 3*C)*x)/(2*a) - ((A - 2*B + 2*C)*Sin[c + d*x])/(a*d) + ((2*A - 2*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, ((B - C)*x)/a + (C*Sin[c + d*x])/(a*d) + ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, (C*x)/a + (A*ArcTanh[Sin[c + d*x]])/(a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 4, -(((A - B)*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*A - B + C)*Tan[c + d*x])/(a*d) - ((A - B + C)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 5, ((3*A - 2*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((2*A - 2*B + C)*Tan[c + d*x])/(a*d) + ((3*A - 2*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 6, -(((3*A - 3*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + (2*(4*A - 3*B + 3*C)*Tan[c + d*x])/(3*a*d) - ((3*A - 3*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((4*A - 3*B + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 5, -(((4*A - 7*B + 10*C)*x)/(2*a^2)) + (2*(5*A - 8*B + 12*C)*Sin[c + d*x])/(3*a^2*d) - ((4*A - 7*B + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + ((5*A - 8*B + 12*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((4*A - 7*B + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 4, ((2*A - 4*B + 7*C)*x)/(2*a^2) - (2*(2*A - 5*B + 8*C)*Sin[c + d*x])/(3*a^2*d) + ((2*A - 4*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((2*A - 5*B + 8*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 4, ((B - 2*C)*x)/a^2 + ((A - B + 4*C)*Sin[c + d*x])/(3*a^2*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((B - 2*C)*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 3, (C*x)/a^2 + ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((A + 2*B - 5*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 3, (A*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((4*A - B - 2*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 5, -(((2*A - B)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + ((10*A - 4*B + C)*Tan[c + d*x])/(3*a^2*d) - ((A - B + C)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((2*A - B)*Tan[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 6, ((7*A - 4*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*(8*A - 5*B + 2*C)*Tan[c + d*x])/(3*a^2*d) + ((7*A - 4*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((8*A - 5*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 7, -(((10*A - 7*B + 4*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d)) + (2*(12*A - 8*B + 5*C)*Tan[c + d*x])/(3*a^2*d) - ((10*A - 7*B + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + ((12*A - 8*B + 5*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((10*A - 7*B + 4*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 6, -(((6*A - 13*B + 23*C)*x)/(2*a^3)) + (8*(9*A - 19*B + 34*C)*Sin[c + d*x])/(15*a^3*d) - ((6*A - 13*B + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + (4*(9*A - 19*B + 34*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^3*d) - ((A - B + C)*Cos[c + d*x]^5*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((3*A - 8*B + 13*C)*Cos[c + d*x]^4*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((6*A - 13*B + 23*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 5, ((2*A - 6*B + 13*C)*x)/(2*a^3) - (2*(11*A - 36*B + 76*C)*Sin[c + d*x])/(15*a^3*d) + ((2*A - 6*B + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((A - 6*B + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((11*A - 36*B + 76*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 5, ((B - 3*C)*x)/a^3 + ((2*A - 7*B + 27*C)*Sin[c + d*x])/(15*a^3*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A + 4*B - 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((B - 3*C)*Sin[c + d*x])/(a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 4, (C*x)/a^3 - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((3*A + 2*B - 7*C)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((3*A + 2*B - 22*C)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 3, ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*A + 3*B - 8*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*A + 3*B + 7*C)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 4, (A*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*A - 2*B - 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((22*A - 2*B - 3*C)*Sin[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 6, -(((3*A - B)*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (2*(36*A - 11*B + C)*Tan[c + d*x])/(15*a^3*d) - ((A - B + C)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*A - 4*B - C)*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((3*A - B)*Tan[c + d*x])/(a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 7, ((13*A - 6*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(76*A - 36*B + 11*C)*Tan[c + d*x])/(15*a^3*d) + ((13*A - 6*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*A - 6*B + C)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((76*A - 36*B + 11*C)*Sec[c + d*x]*Tan[c + d*x])/(15*a^2*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 8, -(((23*A - 13*B + 6*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d)) + (8*(34*A - 19*B + 9*C)*Tan[c + d*x])/(15*a^3*d) - ((23*A - 13*B + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) + (4*(34*A - 19*B + 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a^3*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((13*A - 8*B + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((23*A - 13*B + 6*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 6, ((2*A - 8*B + 21*C)*x)/(2*a^4) - (8*(20*A - 83*B + 216*C)*Sin[c + d*x])/(105*a^4*d) + ((2*A - 8*B + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((A - B + C)*Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((B - 2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3) - ((10*A - 52*B + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - (4*(20*A - 83*B + 216*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 6, ((B - 4*C)*x)/a^4 + ((6*A - 55*B + 244*C)*Sin[c + d*x])/(105*a^4*d) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((2*A + 5*B - 12*C)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((3*A + 25*B - 88*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - ((B - 4*C)*Sin[c + d*x])/(a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 5, (C*x)/a^4 - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((4*A + 3*B - 10*C)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((8*A + 6*B - 55*C)*Cos[c + d*x]*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + (2*(4*A + 3*B - 80*C)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 4, -(((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4)) + ((6*A + B - 8*C)*Cos[c + d*x]*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((A - B - 6*C)*Sin[c + d*x])/(21*a^2*d*(a + a*Cos[c + d*x])^2) + ((8*A + 13*B + 36*C)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 4, ((A - B + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A + 4*B - 11*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((6*A + 8*B + 13*C)*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) + ((6*A + 8*B + 13*C)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 5, (A*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((A - B + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((10*A - 3*B - 4*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - ((55*A - 6*B - 8*C)*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - (2*(80*A - 3*B - 4*C)*Sin[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 7, -(((4*A - B)*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (2*(332*A - 80*B + 3*C)*Tan[c + d*x])/(105*a^4*d) - ((A - B + C)*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((12*A - 5*B - 2*C)*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - ((88*A - 25*B - 3*C)*Tan[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - ((4*A - B)*Tan[c + d*x])/(a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 8, ((21*A - 8*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (8*(216*A - 83*B + 20*C)*Tan[c + d*x])/(105*a^4*d) + ((21*A - 8*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((2*A - B)*Sec[c + d*x]*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3) - ((129*A - 52*B + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - (4*(216*A - 83*B + 20*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^3*d*(a + a*Cos[c + d*x]))}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 9, -(((44*A - 21*B + 8*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d)) + (8*(454*A - 216*B + 83*C)*Tan[c + d*x])/(105*a^4*d) - ((44*A - 21*B + 8*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) + (4*(454*A - 216*B + 83*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((16*A - 9*B + 2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) - ((178*A - 87*B + 31*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^2) - ((44*A - 21*B + 8*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^3*d*(a + a*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 6, (4*a*(99*A + 88*B + 80*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(99*A + 88*B + 80*C)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(11*B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (8*(99*A + 88*B + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*C*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (4*(99*A + 88*B + 80*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*a*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 5, (2*a*(21*A + 18*B + 16*C)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(9*B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) - (4*(21*A + 18*B + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d) + (2*(21*A + 18*B + 16*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 4, (2*a*(35*A + 49*B + 27*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A - 14*B + 18*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (2*(7*B + C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*a*(15*A + 5*B + 7*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*B - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*Sqrt[2]*a*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*B + C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 3, (Sqrt[2]*a*(A + 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) - (a*(A - 2*C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 3, (a*(3*A + 4*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(A + 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 4, (a*(5*A + 6*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(5*A + 6*B + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(A + 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 5, (a*(35*A + 40*B + 48*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(32*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(35*A + 40*B + 48*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(35*A + 40*B + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(A + 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}


{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 6, (2*a^2*(429*A + 374*B + 336*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(99*A + 110*B + 84*C)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(429*A + 374*B + 336*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a*(11*B + 3*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d) + (2*(429*A + 374*B + 336*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 5, (8*a^2*(63*A + 57*B + 47*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(63*A + 57*B + 47*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(63*A - 18*B + 22*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d) + (2*(3*B + C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(21*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (8*a^2*(35*A + 21*B + 19*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(35*A + 21*B + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*B - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (2*Sqrt[2]*a^2*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(15*A + 20*B + 12*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(5*B + 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (Sqrt[2]*a^2*(3*A + 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(3*A - 6*B - 8*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(3*A - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (a^2*(7*A + 12*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(5*A + 4*B - 8*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(3*A + 4*B)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (a^2*(11*A + 14*B + 24*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(19*A + 30*B + 24*C)*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(A + 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 5, (a^2*(75*A + 88*B + 112*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(32*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(75*A + 88*B + 112*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(39*A + 56*B + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(3*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 6, (a^2*(133*A + 150*B + 176*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(64*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(133*A + 150*B + 176*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(133*A + 150*B + 176*C)*Sec[c + d*x]*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(67*A + 90*B + 80*C)*Sec[c + d*x]^2*Tan[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(3*A + 10*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}


{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 7, (2*a^3*(10439*A + 9230*B + 8368*C)*Sin[c + d*x])/(6435*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2717*A + 2522*B + 2224*C)*Cos[c + d*x]^3*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a^2*(10439*A + 9230*B + 8368*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(45045*d) + (2*a^2*(143*A + 182*B + 136*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d) + (2*a*(10439*A + 9230*B + 8368*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15015*d) + (2*a*(13*B + 5*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 6, (64*a^3*(165*A + 143*B + 125*C)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(165*A + 143*B + 125*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a*(165*A + 143*B + 125*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*(99*A - 22*B + 26*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d) + (2*(11*B + 5*C)*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (64*a^3*(21*A + 15*B + 13*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(21*A + 15*B + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(21*A + 15*B + 13*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (2*(9*B - 2*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (2*Sqrt[2]*a^3*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(245*A + 224*B + 160*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(35*A + 56*B + 40*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(7*B + 5*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (Sqrt[2]*a^3*(5*A + 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(15*A + 70*B + 64*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(15*A - 10*B - 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (a*(5*A - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Tan[c + d*x])/d}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (a^3*(19*A + 20*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - (a^3*(27*A - 12*B - 56*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(21*A + 12*B - 8*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (a*(5*A + 4*B)*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (a^3*(25*A + 38*B + 40*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - (a^3*(49*A + 54*B - 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(31*A + 42*B + 24*C)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (a*(5*A + 6*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (a^3*(163*A + 200*B + 304*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(32*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(299*A + 392*B + 432*C)*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(17*A + 24*B + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d) + (a*(5*A + 8*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)}
{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 6, (a^3*(283*A + 326*B + 400*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(64*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(283*A + 326*B + 400*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(787*A + 950*B + 1040*C)*Sec[c + d*x]*Tan[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(79*A + 110*B + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(240*d) + (a*(A + 2*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)}
{Sec[c + d*x]^7*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 7, (a^3*(1015*A + 1132*B + 1304*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(256*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1015*A + 1132*B + 1304*C)*Tan[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1015*A + 1132*B + 1304*C)*Sec[c + d*x]*Tan[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(545*A + 628*B + 680*C)*Sec[c + d*x]^2*Tan[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(115*A + 156*B + 120*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(480*d) + (a*(5*A + 12*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(60*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 6, -((2*(A - B + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + (4*(147*A - 111*B + 143*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(21*A - 3*B + 19*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(9*B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(21*A - 93*B + 29*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 5, (2*(A - B + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) - (4*(35*A - 49*B + 37*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(7*B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A - 7*B + 31*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 4, -((2*(A - B + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + (2*(15*A - 10*B + 14*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*B - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 3, (2*(A - B + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*(3*B - 2*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 4, -((2*(A - B + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + (2*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 4, (2*(A - B + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) - (Sqrt[2]*(A - 2*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) + (A*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 5, -((2*(A - B + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + ((7*A - 4*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 6, (2*(A - B + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]]) - ((9*A - 14*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(4*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) + ((7*A - 2*B + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 7, -((2*(A - B + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(d*Sqrt[a + a*Cos[c + d*x]])) + ((107*A - 72*B + 112*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(32*Sqrt[2]*d*Sqrt[a + a*Cos[c + d*x]]) - ((21*A - 56*B + 16*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + ((43*A - 8*B + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 6, ((11*A - 15*B + 19*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((455*A - 651*B + 799*C)*Sin[c + d*x])/(105*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((35*A - 63*B + 67*C)*Cos[c + d*x]^2*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((7*A - 7*B + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((245*A - 273*B + 397*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(210*a^2*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 5, -(((7*A - 11*B + 15*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((45*A - 65*B + 93*C)*Sin[c + d*x])/(15*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((5*A - 5*B + 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((15*A - 35*B + 39*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(30*a^2*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 4, ((3*A - 7*B + 11*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((3*A - 9*B + 13*C)*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((3*A - 3*B + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 3, ((A + 3*B - 7*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*C*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 4, -(((5*A - B - 3*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])) + (2*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 5, ((9*A - 5*B + C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - (Sqrt[2]*(3*A - 2*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A - B + C)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 6, -(((13*A - 9*B + 5*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])) + ((19*A - 12*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*Sqrt[2]*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((7*A - 6*B + 2*C)*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((2*A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 7, ((17*A - 13*B + 9*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((47*A - 38*B + 24*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(4*Sqrt[2]*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((21*A - 14*B + 12*C)*Tan[c + d*x])/(8*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((13*A - 12*B + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(12*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - 3*B + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])}


{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 6, -(((75*A - 163*B + 283*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((5*A - 13*B + 21*C)*Cos[c + d*x]^3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((465*A - 985*B + 1729*C)*Sin[c + d*x])/(120*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((45*A - 85*B + 157*C)*Cos[c + d*x]^2*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((195*A - 475*B + 787*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*a^3*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 5, ((19*A - 75*B + 163*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((A - 9*B + 17*C)*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((21*A - 93*B + 197*C)*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((15*A - 39*B + 95*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)}
{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 4, ((5*A + 19*B - 75*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A + 5*B - 13*C)*Cos[c + d*x]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((A - 9*B + 49*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 3, ((3*A + 5*B + 19*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A + 5*B - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 5, -(((43*A - 3*B - 5*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])) + (2*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((11*A - 3*B - 5*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 6, ((115*A - 43*B + 3*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - (Sqrt[2]*(5*A - 2*B)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((15*A - 7*B - C)*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((35*A - 11*B + 3*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 7, -(((219*A - 115*B + 43*C)*ArcTanh[Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])) + ((39*A - 20*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(1/2)*(c + d*x)]]*Cos[(1/2)*(c + d*x)])/(2*Sqrt[2]*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((63*A - 35*B + 11*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((19*A - 11*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((31*A - 15*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 6, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 5, (2*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 4, (2*B*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 4, -((2*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/d + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 5, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2), x, 6, -((2*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 8, (2*a*(9*A + 7*B + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*a*(11*A + 11*B + 9*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (10*a*(11*A + 11*B + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(9*A + 7*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(11*A + 11*B + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*a*(11*B + 2*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 7, (2*a*(9*A + 9*B + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*a*(7*A + 5*B + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(7*A + 5*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*A + 9*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(9*B + 2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 6, (2*a*(5*A + 3*B + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(7*A + 7*B + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(7*A + 7*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(7*B + 2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 5, (2*a*(5*A + 5*B + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(3*A + B + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(5*B + 2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 5, -((2*a*(A - B - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*(3*A + 3*B + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*a*(3*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 5, -((2*a*(A + B - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*(A + 3*B + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(2*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 6, -((2*a*(3*A + 5*B + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(A + B + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(2*A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*B + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 7, -((2*a*(3*A + 3*B + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(5*A + 7*B + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(2*A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*a*(5*A + 7*B + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 3*B + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 8, (4*a^2*(9*A + 8*B + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (4*a^2*(66*A + 55*B + 50*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (4*a^2*(66*A + 55*B + 50*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(9*A + 8*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(99*A + 121*B + 89*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*a*(11*B + 4*C)*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 7, (4*a^2*(12*A + 9*B + 8*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^2*(7*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*(9*B + 4*C)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(63*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, (4*a^2*(5*A + 4*B + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^2*(35*A + 49*B + 33*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(7*B + 4*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, (4*a^2*(5*B + 4*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^2*(3*A + 2*B + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*a^2*(15*A - 5*B - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*a*(5*A - C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, -((4*a^2*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (4*a^2*(2*A + 3*B + 2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*a^2*(5*A + 3*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*(4*A + 3*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, -((4*a^2*(4*A + 5*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^2*(A + 2*B + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(17*A + 25*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*(4*A + 5*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 7, -((4*a^2*(3*A + 4*B + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^2*(6*A + 7*B + 14*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^2*(33*A + 49*B + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^2*(3*A + 4*B + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*(4*A + 7*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 8, -((4*a^2*(8*A + 9*B + 12*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (4*a^2*(5*A + 6*B + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^2*(5*A + 6*B + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^2*(8*A + 9*B + 12*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*(4*A + 9*B)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 9, (4*a^3*(221*A + 195*B + 175*C)*EllipticE[(1/2)*(c + d*x), 2])/(195*d) + (4*a^3*(121*A + 105*B + 95*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (4*a^3*(121*A + 105*B + 95*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(221*A + 195*B + 175*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(585*d) + (20*a^3*(286*A + 273*B + 236*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*a^2*(143*A + 195*B + 145*C)*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(1287*d) + (2*a*(13*B + 6*C)*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 8, (4*a^3*(21*A + 17*B + 15*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (4*a^3*(143*A + 121*B + 105*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d) + (2*a^2*(99*A + 143*B + 105*C)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(693*d) + (2*a*(11*B + 6*C)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, (4*a^3*(27*A + 21*B + 17*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^3*(42*A + 41*B + 32*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a^2*(63*A + 99*B + 73*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(315*d) + (2*a*(3*B + 2*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, (4*a^3*(5*A + 9*B + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) - (4*a^3*(35*A - 42*B - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (2*a^2*(35*A - 7*B - 11*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d) - (2*a*(7*A - C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, -((4*a^3*(5*A - 5*B - 9*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^3*(5*A + 5*B + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (4*a^3*(20*A + 5*B - 6*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*a^2*(35*A + 15*B - 3*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (2*a*(2*A + B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, -((4*a^3*(9*A + 5*B - 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^3*(3*A + 5*B + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (4*a^3*(21*A + 20*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a^2*(33*A + 35*B + 15*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*(6*A + 5*B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, -((4*a^3*(7*A + 9*B + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^3*(13*A + 21*B + 35*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^3*(106*A + 147*B + 140*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(7*A + 9*B + 5*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*(6*A + 7*B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 8, -((4*a^3*(17*A + 21*B + 27*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (4*a^3*(11*A + 13*B + 21*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^3*(32*A + 41*B + 42*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(17*A + 21*B + 27*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(73*A + 99*B + 63*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*a*(2*A + 3*B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}
{Cos[c + d*x]^(-13/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 9, -((4*a^3*(15*A + 17*B + 21*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (4*a^3*(105*A + 121*B + 143*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(5/2)) + (4*a^3*(105*A + 121*B + 143*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(3/2)) + (4*a^3*(15*A + 17*B + 21*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(105*A + 143*B + 99*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)) + (2*a*(6*A + 11*B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 7, -((3*(5*A - 7*B + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a*d)) + (5*(7*A - 7*B + 9*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*a*d) + (5*(7*A - 7*B + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((5*A - 7*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((7*A - 7*B + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 6, (3*(5*A - 5*B + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a*d) - ((3*A - 5*B + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) - ((3*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((5*A - 5*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 5, -(((A - 3*B + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) + ((3*A - 3*B + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + ((3*A - 3*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 4, ((A - B + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(a*d) + ((A + B - C)*EllipticF[(1/2)*(c + d*x), 2])/(a*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 5, -(((3*A - B + C)*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - ((A - B - C)*EllipticF[(1/2)*(c + d*x), 2])/(a*d) + ((3*A - B + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 6, ((3*A - 3*B + C)*EllipticE[(1/2)*(c + d*x), 2])/(a*d) + ((5*A - 3*B + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + ((5*A - 3*B + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - ((3*A - 3*B + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 7, -((3*(7*A - 5*B + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a*d)) - ((5*A - 5*B + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + ((7*A - 5*B + 5*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((5*A - 5*B + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(7*A - 5*B + 5*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 7, ((20*A - 35*B + 56*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a^2*d) - (5*(A - 2*B + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (5*(A - 2*B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + ((20*A - 35*B + 56*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((A - 2*B + 3*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 6, -(((A - 4*B + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + ((2*A - 5*B + 10*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + ((2*A - 5*B + 10*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((A - 4*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 5, -(((B - 4*C)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + ((A + 2*B - 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((A + 2*B - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 5, ((A - C)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + ((2*A + B + 2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 6, -(((4*A - B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) - ((5*A - 2*B - C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + ((4*A - B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((5*A - 2*B - C)*Sin[c + d*x])/(3*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 7, ((7*A - 4*B + C)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + ((10*A - 5*B + 2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + ((10*A - 5*B + 2*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((7*A - 4*B + C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((7*A - 4*B + C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}


{Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 8, (7*(7*A - 17*B + 33*C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) - ((13*A - 33*B + 63*C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((13*A - 33*B + 63*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + (7*(7*A - 17*B + 33*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) - ((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - 7*B + 12*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - 33*B + 63*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 7, -(((9*A - 49*B + 119*C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + ((3*A - 13*B + 33*C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + ((3*A - 13*B + 33*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((B - 2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((9*A - 49*B + 119*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 6, -(((A + 9*B - 49*C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + ((A + 3*B - 13*C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*A + 3*B - 8*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A + 3*B - 13*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 6, ((A - B - 9*C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + ((A + B + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*A + B - 6*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - B - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 6, ((9*A + B - C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + ((3*A + B + C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((6*A - B - 4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((9*A + B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 7, -(((49*A - 9*B - C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) - ((13*A - 3*B - C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + ((49*A - 9*B - C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B - 2*C)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B - C)*Sin[c + d*x])/(6*a^2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 8, ((119*A - 49*B + 9*C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + ((33*A - 13*B + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + ((33*A - 13*B + 3*C)*Sin[c + d*x])/(6*a^3*d*Cos[c + d*x]^(3/2)) - ((119*A - 49*B + 9*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((119*A - 49*B + 9*C)*Sin[c + d*x])/(30*a^2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^(n/2)*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 5, (Sqrt[a]*(48*A + 40*B + 35*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*d) + (a*(48*A + 40*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(48*A + 40*B + 35*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(8*B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 4, (Sqrt[a]*(8*A + 6*B + 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) + (a*(8*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(6*B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 3, (Sqrt[a]*(8*A + 4*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) + (a*(4*B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 3, (Sqrt[a]*(2*B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d - (a*(2*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 3, (2*Sqrt[a]*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a*(A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2), x, 3, (2*a*(A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(8*A + 10*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(9/2), x, 4, (2*a*(A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 28*B + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(24*A + 28*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(11/2), x, 5, (2*a*(A + 9*B)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}


{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 6, (a^(3/2)*(176*A + 150*B + 133*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(128*d) + (a^2*(176*A + 150*B + 133*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(176*A + 150*B + 133*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 90*B + 67*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(10*B + 3*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 5, (a^(3/2)*(112*A + 88*B + 75*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*d) + (a^2*(112*A + 88*B + 75*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(48*A + 56*B + 39*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(8*B + 3*C)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 4, (a^(3/2)*(24*A + 14*B + 11*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) + (a^2*(24*A + 30*B + 19*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(2*B + C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 4, (a^(3/2)*(8*A + 12*B + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) - (a^2*(8*A - 4*B - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(4*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 4, (a^(3/2)*(2*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d - (a^2*(8*A + 6*B - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2), x, 4, (2*a^(3/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a^2*(12*A + 20*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(9/2), x, 4, (8*a^2*(19*A + 21*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(19*A + 21*B + 35*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(3*A + 7*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(11/2), x, 5, (2*a^2*(52*A + 72*B + 63*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 156*B + 189*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(136*A + 156*B + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(13/2), x, 6, (2*a^2*(84*A + 110*B + 99*C)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))}


{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 7, (a^(5/2)*(1304*A + 1132*B + 1015*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(512*d) + (a^3*(1304*A + 1132*B + 1015*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1304*A + 1132*B + 1015*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(680*A + 628*B + 545*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(120*A + 156*B + 115*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(480*d) + (a*(12*B + 5*C)*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(60*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 6, (a^(5/2)*(400*A + 326*B + 283*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(128*d) + (a^3*(400*A + 326*B + 283*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1040*A + 950*B + 787*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 110*B + 79*C)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d) + (a*(2*B + C)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 5, (a^(5/2)*(304*A + 200*B + 163*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*d) + (a^3*(432*A + 392*B + 299*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(16*A + 24*B + 17*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + (a*(8*B + 5*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 5, (a^(5/2)*(40*A + 38*B + 25*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*d) - (a^3*(24*A - 54*B - 49*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A - 2*B - 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (a*(6*A - C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 5, (a^(5/2)*(8*A + 20*B + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*d) - (a^3*(56*A + 12*B - 27*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A + 4*B - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*a*(5*A + 3*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2), x, 5, (a^(5/2)*(2*B + 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d - (a^3*(64*A + 70*B + 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 10*B + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*(A + B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(9/2), x, 5, (2*a^(5/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a^3*(160*A + 224*B + 245*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(40*A + 56*B + 35*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 7*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(11/2), x, 5, (64*a^3*(13*A + 15*B + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(13*A + 15*B + 21*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)) + (2*a*(13*A + 15*B + 21*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(5*A + 9*B)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(13/2), x, 6, (2*a^3*(1160*A + 1364*B + 1485*C)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2840*A + 3212*B + 3795*C)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(2840*A + 3212*B + 3795*C)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 44*B + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (2*a*(5*A + 11*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(15/2), x, 7, (2*a^3*(2224*A + 2522*B + 2717*C)*Sin[c + d*x])/(9009*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(15015*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(45045*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 182*B + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d*Cos[c + d*x]^(9/2)) + (2*a*(5*A + 13*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Cos[c + d*x]^(11/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Cos[c + d*x]^(13/2))}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]], x, 6, -(((8*A - 14*B + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((8*A - 2*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + ((6*B - C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{(Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]], x, 5, ((8*A - 4*B + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((4*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(1/2)*Sqrt[a + a*Cos[c + d*x]]), x, 4, ((2*B - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]), x, 4, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]), x, 3, (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]), x, 4, -((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(13*A - 5*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + a*Cos[c + d*x]]), x, 5, (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A - 7*B + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(43*A - 91*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}

{(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]], x, 5, ((8*a*A - 4*A*b - 4*a*B + 7*b*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*Sqrt[a]*d) - (Sqrt[2]*(a*A - A*b - a*B + b*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((4*A*b + 4*a*B - b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (b*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])}


{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2), x, 6, ((8*A - 12*B + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((2*A - 6*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B + 2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{(Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2), x, 5, ((2*B - 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(3/2)*d) + ((A - 5*B + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((A - B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^(3/2)), x, 4, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(3/2)*d) + ((3*A + B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)), x, 3, -(((7*A - 3*B - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B + C)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)), x, 4, ((11*A - 7*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B + 3*C)*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((19*A - 15*B + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])^(3/2)), x, 5, -(((15*A - 11*B + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 5*B + 5*C)*Sin[c + d*x])/(10*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 35*B + 15*C)*Sin[c + d*x])/(30*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + ((147*A - 95*B + 75*C)*Sin[c + d*x])/(30*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2), x, 6, ((2*B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((A + 7*B - 15*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A - 11*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{(Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2), x, 5, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A + 3*B - 11*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^(5/2)), x, 3, ((19*A + 5*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((9*A - B - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)), x, 4, -(((75*A - 19*B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B + C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B - 3*C)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B + C)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)), x, 5, ((163*A - 75*B + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B + C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((95*A - 39*B + 15*C)*Sin[c + d*x])/(48*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((299*A - 147*B + 27*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^(n/3)*)
(**)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(2/3), x, 3, (1/80)*a*(40*A + 16*B + 19*C)*Int[1/(a + a*Cos[c + d*x])^(1/3), x] + (3*a*(40*A + 16*B + 19*C)*Sin[c + d*x])/(80*d*(a + a*Cos[c + d*x])^(1/3)) + (3*(8*B - 3*C)*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(40*d) + (3*C*(a + a*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*a*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/3), x, 3, (-(1/28))*a*(28*A + 7*B + 13*C)*Int[1/(a + a*sin[c + Pi/2 + d*x])^(2/3), x] + (3*a*(28*A + 7*B + 13*C)*Sin[c + d*x])/(28*d*(a + a*Cos[c + d*x])^(2/3)) + (3*(7*B - 3*C)*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(28*d) + (3*C*(a + a*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*a*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/3), x, 2, (1/10)*(10*A - 5*B + 7*C)*Int[1/(a + a*Cos[c + d*x])^(1/3), x] + (3*(5*B - 3*C)*Sin[c + d*x])/(10*d*(a + a*Cos[c + d*x])^(1/3)) + (3*C*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*a*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(2/3), x, 2, (1/4)*(4*A - 8*B + 7*C)*Int[1/(a + a*sin[c + Pi/2 + d*x])^(2/3), x] + (3*(4*B - 3*C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(2/3)) + (3*C*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*a*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Sec[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>=0*)


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, -((2*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, (-2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 5, (-2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 5, (2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Sec[c + d*x]], x, 6, (2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 7, (6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, -((2*a*(3*A + 3*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(5*A + 7*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(3*A + 3*B + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*A + 7*B + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*(2*A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, -((2*a*(3*A + 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*A + 5*B + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(2*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, (-2*a*(A + B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(2*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 6, (-2*a*(A - B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + 3*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a*(3*A - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 6, (2*a*(5*A + 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(5*B + 2*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Sec[c + d*x]], x, 7, (2*a*(5*A + 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 7*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(7*B + 2*C)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 7*B + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 8, (2*a*(9*A + 9*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*A + 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(9*B + 2*C)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^(5/2)) + (2*a*(9*A + 9*B + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*B + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 9, -((4*a^2*(8*A + 9*B + 12*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d)) + (4*a^2*(5*A + 6*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(8*A + 9*B + 12*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*a*(4*A + 9*B)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, -((4*a^2*(3*A + 4*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^2*(6*A + 7*B + 14*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(3*A + 4*B + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(33*A + 49*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*(4*A + 7*B)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, -((4*a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^2*(A + 2*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(17*A + 25*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(4*A + 5*B)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 7, (-4*a^2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(2*A + 3*B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(5*A + 3*B - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*(4*A + 3*B)*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 7, (4*a^2*(5*B + 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(3*A + 2*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(15*A - 5*B - 7*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*a*(5*A - C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 7, (4*a^2*(5*A + 4*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(35*A + 49*B + 33*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*(7*B + 4*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Sec[c + d*x]], x, 8, (4*a^2*(12*A + 9*B + 8*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (2*a*(9*B + 4*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (4*a^2*(7*A + 6*B + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 9, (4*a^2*(9*A + 8*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(66*A + 55*B + 50*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(99*A + 121*B + 89*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*a*(11*B + 4*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (4*a^2*(9*A + 8*B + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (4*a^2*(66*A + 55*B + 50*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 10, -((4*a^3*(15*A + 17*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d)) + (4*a^3*(105*A + 121*B + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(15*A + 17*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d) + (2*a^2*(105*A + 143*B + 99*C)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d) + (2*a*(6*A + 11*B)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 9, -((4*a^3*(17*A + 21*B + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d)) + (4*a^3*(11*A + 13*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(17*A + 21*B + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(32*A + 41*B + 42*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a^2*(73*A + 99*B + 63*C)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*a*(2*A + 3*B)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, -((4*a^3*(7*A + 9*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^3*(13*A + 21*B + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(106*A + 147*B + 140*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a^2*(7*A + 9*B + 5*C)*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*(6*A + 7*B)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 8, -((4*a^3*(9*A + 5*B - 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^3*(3*A + 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(21*A + 20*B + 5*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(33*A + 35*B + 15*C)*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(6*A + 5*B)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 8, (-4*a^3*(5*A - 5*B - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(20*A + 5*B - 6*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*a^2*(35*A + 15*B - 3*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*(2*A + B)*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 8, (4*a^3*(5*A + 9*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a^3*(35*A - 42*B - 41*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) - (2*a^2*(35*A - 7*B - 11*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]) - (2*a*(7*A - C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 8, (4*a^3*(27*A + 21*B + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(42*A + 41*B + 32*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(63*A + 99*B + 73*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]) + (2*a*(3*B + 2*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Sec[c + d*x]], x, 9, (4*a^3*(21*A + 17*B + 15*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*Sin[c + d*x])/(1155*d*Sec[c + d*x]^(3/2)) + (2*a^2*(99*A + 143*B + 105*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(693*d*Sec[c + d*x]^(3/2)) + (2*a*(11*B + 6*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(99*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(3/2)) + (4*a^3*(143*A + 121*B + 105*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 10, (4*a^3*(221*A + 195*B + 175*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (4*a^3*(121*A + 105*B + 95*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (20*a^3*(286*A + 273*B + 236*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*a^2*(143*A + 195*B + 145*C)*(a + a*Cos[c + d*x])*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(5/2)) + (2*a*(13*B + 6*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(143*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(5/2)) + (4*a^3*(221*A + 195*B + 175*C)*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (4*a^3*(121*A + 105*B + 95*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2)/(a + a*Cos[c + d*x]), x, 8, -((3*(7*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a*d)) - ((5*A - 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(7*A - 5*B + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) - ((5*A - 5*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((7*A - 5*B + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x]), x, 7, ((3*A - 3*B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((5*A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((3*A - 3*B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((5*A - 3*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x]), x, 6, -(((3*A - B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) - ((A - B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x]), x, 5, ((A - B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A + B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]), x, 6, -(((A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((3*A - 3*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A - 3*B + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 7, (3*(5*A - 5*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((3*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((5*A - 5*B + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - ((3*A - 5*B + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)), x, 8, (-3*(5*A - 7*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(7*A - 7*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)) + ((7*A - 7*B + 9*C)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - ((5*A - 7*B + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(7*A - 7*B + 9*C)*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]])}


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^2, x, 8, ((7*A - 4*B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((10*A - 5*B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((7*A - 4*B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((10*A - 5*B + 2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((7*A - 4*B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^2, x, 7, -(((4*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) - ((5*A - 2*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((4*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((5*A - 2*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x]))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x])^2, x, 6, ((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((2*A + B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A - C)*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]), x, 6, -(((B - 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((A + 2*B - 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A + 2*B - 5*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)), x, 7, -(((A - 4*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((2*A - 5*B + 10*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((A - 4*B + 7*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((2*A - 5*B + 10*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)), x, 8, ((20*A - 35*B + 56*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(A - 2*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((A - 2*B + 3*C)*Sin[c + d*x])/(a*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((20*A - 35*B + 56*C)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(A - 2*B + 3*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])}


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^3, x, 9, ((119*A - 49*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((33*A - 13*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((119*A - 49*B + 9*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((33*A - 13*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a^3*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((119*A - 49*B + 9*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x]))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^3, x, 8, -((49*A - 9*B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 3*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((49*A - 9*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B - 2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x]))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x])^3, x, 7, ((9*A + B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]) - ((6*A - B - 4*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((9*A + B - C)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]), x, 7, ((A - B - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)) + ((4*A + B - 6*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A - B - 9*C)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)), x, 7, -((A + 9*B - 49*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*B - 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)) + ((2*A + 3*B - 8*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A + 3*B - 13*C)*Sin[c + d*x])/(6*a^2*d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)), x, 8, -((9*A - 49*B + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A - 13*B + 33*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)) + ((B - 2*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((9*A - 49*B + 119*C)*Sin[c + d*x])/(30*a^2*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A - 13*B + 33*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)), x, 9, (7*(7*A - 17*B + 33*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 33*B + 63*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)) - ((2*A - 7*B + 12*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((13*A - 33*B + 63*C)*Sin[c + d*x])/(10*a^2*d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (7*(7*A - 17*B + 33*C)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((13*A - 33*B + 63*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Sec[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+a Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 5, (16*a*(16*A + 18*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 18*B + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 18*B + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 4, (4*a*(24*A + 28*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 28*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 3, (2*a*(8*A + 10*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 4, (2*Sqrt[a]*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 4, (Sqrt[a]*(2*B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a*(2*A - C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2), x, 4, (Sqrt[a]*(8*A + 4*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*(4*B + C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(1/2), x, 5, (Sqrt[a]*(8*A + 6*B + 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*(6*B + C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (a*(8*A + 6*B + 5*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 6, (Sqrt[a]*(48*A + 40*B + 35*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a*(8*B + C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (a*(48*A + 40*B + 35*C)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(48*A + 40*B + 35*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 6, (16*a^2*(336*A + 374*B + 429*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(336*A + 374*B + 429*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(336*A + 374*B + 429*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(84*A + 110*B + 99*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 5, (4*a^2*(136*A + 156*B + 189*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 156*B + 189*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 72*B + 63*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 4, (8*a^2*(19*A + 21*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(19*A + 21*B + 35*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(3*A + 7*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 5, (2*a^(3/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(12*A + 20*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 5, (a^(3/2)*(2*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^2*(8*A + 6*B - 3*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*(A + B)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 5, (a^(3/2)*(8*A + 12*B + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^2*(8*A - 4*B - 5*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a*(4*A - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2), x, 5, (a^(3/2)*(24*A + 14*B + 11*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(24*A + 30*B + 19*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*(2*B + C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(1/2), x, 6, (a^(3/2)*(112*A + 88*B + 75*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(48*A + 56*B + 39*C)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(8*B + 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)) + (a^2*(112*A + 88*B + 75*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 7, (a^(3/2)*(176*A + 150*B + 133*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^2*(80*A + 90*B + 67*C)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a*(10*B + 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(5/2)) + (a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(15/2), x, 7, (16*a^3*(8368*A + 9230*B + 10439*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 9230*B + 10439*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 9230*B + 10439*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2224*A + 2522*B + 2717*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 182*B + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(1287*d) + (2*a*(5*A + 13*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(143*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(13/2)*Sin[c + d*x])/(13*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 6, (4*a^3*(2840*A + 3212*B + 3795*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2840*A + 3212*B + 3795*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(1160*A + 1364*B + 1485*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 44*B + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (2*a*(5*A + 11*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 5, (64*a^3*(13*A + 15*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(13*A + 15*B + 21*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*a*(13*A + 15*B + 21*C)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*(5*A + 9*B)*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 6, (2*a^(5/2)*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(160*A + 224*B + 245*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(40*A + 56*B + 35*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*(5*A + 7*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 6, (a^(5/2)*(2*B + 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^3*(64*A + 70*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*(8*A + 10*B + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(A + B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, (a^(5/2)*(8*A + 20*B + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^3*(56*A + 12*B - 27*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A + 4*B - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*a*(5*A + 3*B)*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 6, (a^(5/2)*(40*A + 38*B + 25*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) - (a^3*(24*A - 54*B - 49*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A - 2*B - 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) - (a*(6*A - C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2), x, 6, (a^(5/2)*(304*A + 200*B + 163*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(432*A + 392*B + 299*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*(16*A + 24*B + 17*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + (a*(8*B + 5*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(1/2), x, 7, (a^(5/2)*(400*A + 326*B + 283*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 950*B + 787*C)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(80*A + 110*B + 79*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d*Sec[c + d*x]^(3/2)) + (a*(2*B + C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (a^3*(400*A + 326*B + 283*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 8, (a^(5/2)*(1304*A + 1132*B + 1015*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(512*d) + (a^3*(680*A + 628*B + 545*C)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*(120*A + 156*B + 115*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(480*d*Sec[c + d*x]^(5/2)) + (a*(12*B + 5*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(60*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Sec[c + d*x]^(5/2)) + (a^3*(1304*A + 1132*B + 1015*C)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^3*(1304*A + 1132*B + 1015*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2)/Sqrt[a + a*Cos[c + d*x]], x, 7, -((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(257*A - 129*B + 273*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(29*A - 93*B + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(19*A - 3*B + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2)/Sqrt[a + a*Cos[c + d*x]], x, 6, (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A - 91*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A - 7*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2)/Sqrt[a + a*Cos[c + d*x]], x, 5, -((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A - 5*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2)/Sqrt[a + a*Cos[c + d*x]], x, 4, (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2)/Sqrt[a + a*Cos[c + d*x]], x, 5, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2)/Sqrt[a + a*Cos[c + d*x]], x, 5, ((2*B - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(1/2)), x, 6, ((8*A - 4*B + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((4*B - C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 7, -(((8*A - 14*B + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + ((6*B - C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((8*A - 2*B + 7*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}

{((a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]], x, 5, ((2*A*b + 2*a*B - b*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(a*A - A*b - a*B + b*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (b*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^(3/2), x, 7, ((19*A - 15*B + 11*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((1201*A - 1029*B + 665*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((397*A - 273*B + 245*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((67*A - 63*B + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((11*A - 7*B + 7*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(3/2), x, 6, -(((15*A - 11*B + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) + ((147*A - 95*B + 75*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 35*B + 15*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 5*B + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(3/2), x, 5, ((11*A - 7*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((19*A - 15*B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(3/2), x, 4, -(((7*A - 3*B - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(3/2), x, 5, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((3*A + B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(1/2)), x, 6, ((2*B - 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A - 5*B + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((A - B + 3*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)), x, 7, ((8*A - 12*B + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((A - B + 2*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((2*A - 6*B + 7*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(5/2), x, 7, -(((283*A - 163*B + 75*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) + ((2671*A - 1495*B + 735*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((787*A - 475*B + 195*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((21*A - 13*B + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((157*A - 85*B + 45*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(5/2), x, 6, ((163*A - 75*B + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((299*A - 147*B + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((95*A - 39*B + 15*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(5/2), x, 5, -(((75*A - 19*B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(5/2), x, 4, ((19*A + 5*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((9*A - B - 7*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(1/2)), x, 6, (2*C*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((5*A + 3*B - 11*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 7, ((2*B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((A + 7*B - 15*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((3*A - 11*B + 35*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)), x, 8, ((8*A - 20*B + 39*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(5/2)*d) - ((43*A - 115*B + 219*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)) - ((3*A - 11*B + 19*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((7*A - 15*B + 31*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((11*A - 35*B + 63*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])}
